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Theorem poimirlem4 33045
 Description: Lemma for poimir 33074 connecting the admissible faces on the back face of the (𝑀 + 1)-cube to admissible simplices in the 𝑀-cube. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem4.1 (𝜑𝐾 ∈ ℕ)
poimirlem4.2 (𝜑𝑀 ∈ ℕ0)
poimirlem4.3 (𝜑𝑀 < 𝑁)
Assertion
Ref Expression
poimirlem4 (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
Distinct variable groups:   𝑓,𝑖,𝑗,𝑝,𝑠   𝜑,𝑗   𝑗,𝑀   𝑗,𝑁   𝜑,𝑖,𝑝,𝑠   𝐵,𝑓,𝑖,𝑗,𝑠   𝑓,𝐾,𝑖,𝑗,𝑝,𝑠   𝑓,𝑀,𝑖,𝑝,𝑠   𝑓,𝑁,𝑖,𝑝,𝑠
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑝)

Proof of Theorem poimirlem4
Dummy variables 𝑘 𝑛 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poimir.0 . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
21adantr 481 . . . . . . 7 ((𝜑𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝑁 ∈ ℕ)
3 poimirlem4.1 . . . . . . . 8 (𝜑𝐾 ∈ ℕ)
43adantr 481 . . . . . . 7 ((𝜑𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝐾 ∈ ℕ)
5 poimirlem4.2 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
65adantr 481 . . . . . . 7 ((𝜑𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝑀 ∈ ℕ0)
7 poimirlem4.3 . . . . . . . 8 (𝜑𝑀 < 𝑁)
87adantr 481 . . . . . . 7 ((𝜑𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → 𝑀 < 𝑁)
9 xp1st 7143 . . . . . . . . 9 (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (1st𝑡) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)))
10 elmapi 7823 . . . . . . . . 9 ((1st𝑡) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) → (1st𝑡):(1...𝑀)⟶(0..^𝐾))
119, 10syl 17 . . . . . . . 8 (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (1st𝑡):(1...𝑀)⟶(0..^𝐾))
1211adantl 482 . . . . . . 7 ((𝜑𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (1st𝑡):(1...𝑀)⟶(0..^𝐾))
13 xp2nd 7144 . . . . . . . . 9 (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd𝑡) ∈ {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})
14 fvex 6158 . . . . . . . . . 10 (2nd𝑡) ∈ V
15 f1oeq1 6084 . . . . . . . . . 10 (𝑓 = (2nd𝑡) → (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) ↔ (2nd𝑡):(1...𝑀)–1-1-onto→(1...𝑀)))
1614, 15elab 3333 . . . . . . . . 9 ((2nd𝑡) ∈ {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ↔ (2nd𝑡):(1...𝑀)–1-1-onto→(1...𝑀))
1713, 16sylib 208 . . . . . . . 8 (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd𝑡):(1...𝑀)–1-1-onto→(1...𝑀))
1817adantl 482 . . . . . . 7 ((𝜑𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (2nd𝑡):(1...𝑀)–1-1-onto→(1...𝑀))
192, 4, 6, 8, 12, 18poimirlem3 33044 . . . . . 6 ((𝜑𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 → (⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))))
20 fvex 6158 . . . . . . . . . . . . . . . 16 (1st𝑡) ∈ V
21 snex 4869 . . . . . . . . . . . . . . . 16 {⟨(𝑀 + 1), 0⟩} ∈ V
2220, 21unex 6909 . . . . . . . . . . . . . . 15 ((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∈ V
23 snex 4869 . . . . . . . . . . . . . . . 16 {⟨(𝑀 + 1), (𝑀 + 1)⟩} ∈ V
2414, 23unex 6909 . . . . . . . . . . . . . . 15 ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∈ V
2522, 24op1std 7123 . . . . . . . . . . . . . 14 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (1st𝑠) = ((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}))
2622, 24op2ndd 7124 . . . . . . . . . . . . . . . . 17 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (2nd𝑠) = ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
2726imaeq1d 5424 . . . . . . . . . . . . . . . 16 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((2nd𝑠) “ (1...𝑗)) = (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)))
2827xpeq1d 5098 . . . . . . . . . . . . . . 15 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((2nd𝑠) “ (1...𝑗)) × {1}) = ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}))
2926imaeq1d 5424 . . . . . . . . . . . . . . . 16 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) = (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))))
3029xpeq1d 5098 . . . . . . . . . . . . . . 15 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))
3128, 30uneq12d 3746 . . . . . . . . . . . . . 14 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})))
3225, 31oveq12d 6622 . . . . . . . . . . . . 13 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))))
3332uneq1d 3744 . . . . . . . . . . . 12 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
3433csbeq1d 3521 . . . . . . . . . . 11 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵)
3534eqeq2d 2631 . . . . . . . . . 10 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
3635rexbidv 3045 . . . . . . . . 9 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
3736ralbidv 2980 . . . . . . . 8 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
3825fveq1d 6150 . . . . . . . . 9 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((1st𝑠)‘(𝑀 + 1)) = (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)))
3938eqeq1d 2623 . . . . . . . 8 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((1st𝑠)‘(𝑀 + 1)) = 0 ↔ (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0))
4026fveq1d 6150 . . . . . . . . 9 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((2nd𝑠)‘(𝑀 + 1)) = (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)))
4140eqeq1d 2623 . . . . . . . 8 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1) ↔ (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1)))
4237, 39, 413anbi123d 1396 . . . . . . 7 (𝑠 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))))
4342elrab 3346 . . . . . 6 (⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ↔ (⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∘𝑓 + (((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩})‘(𝑀 + 1)) = 0 ∧ (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})‘(𝑀 + 1)) = (𝑀 + 1))))
4419, 43syl6ibr 242 . . . . 5 ((𝜑𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 → ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}))
4544ralrimiva 2960 . . . 4 (𝜑 → ∀𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 → ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}))
46 fveq2 6148 . . . . . . . . . . 11 (𝑠 = 𝑡 → (1st𝑠) = (1st𝑡))
47 fveq2 6148 . . . . . . . . . . . . . 14 (𝑠 = 𝑡 → (2nd𝑠) = (2nd𝑡))
4847imaeq1d 5424 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → ((2nd𝑠) “ (1...𝑗)) = ((2nd𝑡) “ (1...𝑗)))
4948xpeq1d 5098 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (((2nd𝑠) “ (1...𝑗)) × {1}) = (((2nd𝑡) “ (1...𝑗)) × {1}))
5047imaeq1d 5424 . . . . . . . . . . . . 13 (𝑠 = 𝑡 → ((2nd𝑠) “ ((𝑗 + 1)...𝑀)) = ((2nd𝑡) “ ((𝑗 + 1)...𝑀)))
5150xpeq1d 5098 . . . . . . . . . . . 12 (𝑠 = 𝑡 → (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}) = (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))
5249, 51uneq12d 3746 . . . . . . . . . . 11 (𝑠 = 𝑡 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0})) = ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0})))
5346, 52oveq12d 6622 . . . . . . . . . 10 (𝑠 = 𝑡 → ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) = ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))))
5453uneq1d 3744 . . . . . . . . 9 (𝑠 = 𝑡 → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})))
5554csbeq1d 3521 . . . . . . . 8 (𝑠 = 𝑡(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵)
5655eqeq2d 2631 . . . . . . 7 (𝑠 = 𝑡 → (𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
5756rexbidv 3045 . . . . . 6 (𝑠 = 𝑡 → (∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
5857ralbidv 2980 . . . . 5 (𝑠 = 𝑡 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
5958ralrab 3350 . . . 4 (∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ↔ ∀𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 → ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}))
6045, 59sylibr 224 . . 3 (𝜑 → ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
61 xp1st 7143 . . . . . . . . . . . . . . 15 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (1st𝑘) ∈ ((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))))
62 elmapi 7823 . . . . . . . . . . . . . . 15 ((1st𝑘) ∈ ((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) → (1st𝑘):(1...(𝑀 + 1))⟶(0..^𝐾))
6361, 62syl 17 . . . . . . . . . . . . . 14 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (1st𝑘):(1...(𝑀 + 1))⟶(0..^𝐾))
64 fzssp1 12326 . . . . . . . . . . . . . 14 (1...𝑀) ⊆ (1...(𝑀 + 1))
65 fssres 6027 . . . . . . . . . . . . . 14 (((1st𝑘):(1...(𝑀 + 1))⟶(0..^𝐾) ∧ (1...𝑀) ⊆ (1...(𝑀 + 1))) → ((1st𝑘) ↾ (1...𝑀)):(1...𝑀)⟶(0..^𝐾))
6663, 64, 65sylancl 693 . . . . . . . . . . . . 13 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((1st𝑘) ↾ (1...𝑀)):(1...𝑀)⟶(0..^𝐾))
67 ovex 6632 . . . . . . . . . . . . . 14 (0..^𝐾) ∈ V
68 ovex 6632 . . . . . . . . . . . . . 14 (1...𝑀) ∈ V
6967, 68elmap 7830 . . . . . . . . . . . . 13 (((1st𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) ↔ ((1st𝑘) ↾ (1...𝑀)):(1...𝑀)⟶(0..^𝐾))
7066, 69sylibr 224 . . . . . . . . . . . 12 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((1st𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)))
7170ad2antlr 762 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((1st𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)))
72 xp2nd 7144 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd𝑘) ∈ {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})
73 fvex 6158 . . . . . . . . . . . . . . . . . 18 (2nd𝑘) ∈ V
74 f1oeq1 6084 . . . . . . . . . . . . . . . . . 18 (𝑓 = (2nd𝑘) → (𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ (2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))))
7573, 74elab 3333 . . . . . . . . . . . . . . . . 17 ((2nd𝑘) ∈ {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))} ↔ (2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)))
7672, 75sylib 208 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)))
77 f1of1 6093 . . . . . . . . . . . . . . . 16 ((2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → (2nd𝑘):(1...(𝑀 + 1))–1-1→(1...(𝑀 + 1)))
7876, 77syl 17 . . . . . . . . . . . . . . 15 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd𝑘):(1...(𝑀 + 1))–1-1→(1...(𝑀 + 1)))
79 f1ores 6108 . . . . . . . . . . . . . . 15 (((2nd𝑘):(1...(𝑀 + 1))–1-1→(1...(𝑀 + 1)) ∧ (1...𝑀) ⊆ (1...(𝑀 + 1))) → ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd𝑘) “ (1...𝑀)))
8078, 64, 79sylancl 693 . . . . . . . . . . . . . 14 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd𝑘) “ (1...𝑀)))
8180ad2antlr 762 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd𝑘) “ (1...𝑀)))
82 dff1o3 6100 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) ↔ ((2nd𝑘):(1...(𝑀 + 1))–onto→(1...(𝑀 + 1)) ∧ Fun (2nd𝑘)))
8382simprbi 480 . . . . . . . . . . . . . . . . . 18 ((2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → Fun (2nd𝑘))
84 imadif 5931 . . . . . . . . . . . . . . . . . 18 (Fun (2nd𝑘) → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = (((2nd𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd𝑘) “ {(𝑀 + 1)})))
8576, 83, 843syl 18 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = (((2nd𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd𝑘) “ {(𝑀 + 1)})))
8685ad2antlr 762 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = (((2nd𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd𝑘) “ {(𝑀 + 1)})))
87 f1ofo 6101 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → (2nd𝑘):(1...(𝑀 + 1))–onto→(1...(𝑀 + 1)))
88 foima 6077 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑘):(1...(𝑀 + 1))–onto→(1...(𝑀 + 1)) → ((2nd𝑘) “ (1...(𝑀 + 1))) = (1...(𝑀 + 1)))
8976, 87, 883syl 18 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd𝑘) “ (1...(𝑀 + 1))) = (1...(𝑀 + 1)))
9089ad2antlr 762 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) “ (1...(𝑀 + 1))) = (1...(𝑀 + 1)))
91 f1ofn 6095 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑘):(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1)) → (2nd𝑘) Fn (1...(𝑀 + 1)))
9276, 91syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (2nd𝑘) Fn (1...(𝑀 + 1)))
93 nn0p1nn 11276 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ)
945, 93syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑀 + 1) ∈ ℕ)
95 elfz1end 12313 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 + 1) ∈ ℕ ↔ (𝑀 + 1) ∈ (1...(𝑀 + 1)))
9694, 95sylib 208 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑀 + 1) ∈ (1...(𝑀 + 1)))
97 fnsnfv 6215 . . . . . . . . . . . . . . . . . . 19 (((2nd𝑘) Fn (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (1...(𝑀 + 1))) → {((2nd𝑘)‘(𝑀 + 1))} = ((2nd𝑘) “ {(𝑀 + 1)}))
9892, 96, 97syl2anr 495 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → {((2nd𝑘)‘(𝑀 + 1))} = ((2nd𝑘) “ {(𝑀 + 1)}))
99 sneq 4158 . . . . . . . . . . . . . . . . . 18 (((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → {((2nd𝑘)‘(𝑀 + 1))} = {(𝑀 + 1)})
10098, 99sylan9req 2676 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) “ {(𝑀 + 1)}) = {(𝑀 + 1)})
10190, 100difeq12d 3707 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (((2nd𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd𝑘) “ {(𝑀 + 1)})) = ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}))
10286, 101eqtrd 2655 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}))
103 1z 11351 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℤ
104 nn0uz 11666 . . . . . . . . . . . . . . . . . . . . . . 23 0 = (ℤ‘0)
105 1m1e0 11033 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 − 1) = 0
106105fveq2i 6151 . . . . . . . . . . . . . . . . . . . . . . 23 (ℤ‘(1 − 1)) = (ℤ‘0)
107104, 106eqtr4i 2646 . . . . . . . . . . . . . . . . . . . . . 22 0 = (ℤ‘(1 − 1))
1085, 107syl6eleq 2708 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 ∈ (ℤ‘(1 − 1)))
109 fzsuc2 12340 . . . . . . . . . . . . . . . . . . . . 21 ((1 ∈ ℤ ∧ 𝑀 ∈ (ℤ‘(1 − 1))) → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
110103, 108, 109sylancr 694 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...(𝑀 + 1)) = ((1...𝑀) ∪ {(𝑀 + 1)}))
111110difeq1d 3705 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (((1...𝑀) ∪ {(𝑀 + 1)}) ∖ {(𝑀 + 1)}))
112 difun2 4020 . . . . . . . . . . . . . . . . . . 19 (((1...𝑀) ∪ {(𝑀 + 1)}) ∖ {(𝑀 + 1)}) = ((1...𝑀) ∖ {(𝑀 + 1)})
113111, 112syl6eq 2671 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = ((1...𝑀) ∖ {(𝑀 + 1)}))
1145nn0red 11296 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑀 ∈ ℝ)
115 ltp1 10805 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℝ → 𝑀 < (𝑀 + 1))
116 peano2re 10153 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ)
117 ltnle 10061 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 ∈ ℝ ∧ (𝑀 + 1) ∈ ℝ) → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
118116, 117mpdan 701 . . . . . . . . . . . . . . . . . . . . . 22 (𝑀 ∈ ℝ → (𝑀 < (𝑀 + 1) ↔ ¬ (𝑀 + 1) ≤ 𝑀))
119115, 118mpbid 222 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℝ → ¬ (𝑀 + 1) ≤ 𝑀)
120114, 119syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ¬ (𝑀 + 1) ≤ 𝑀)
121 elfzle2 12287 . . . . . . . . . . . . . . . . . . . 20 ((𝑀 + 1) ∈ (1...𝑀) → (𝑀 + 1) ≤ 𝑀)
122120, 121nsyl 135 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ¬ (𝑀 + 1) ∈ (1...𝑀))
123 difsn 4297 . . . . . . . . . . . . . . . . . . 19 (¬ (𝑀 + 1) ∈ (1...𝑀) → ((1...𝑀) ∖ {(𝑀 + 1)}) = (1...𝑀))
124122, 123syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...𝑀) ∖ {(𝑀 + 1)}) = (1...𝑀))
125113, 124eqtrd 2655 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (1...𝑀))
126125imaeq2d 5425 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd𝑘) “ (1...𝑀)))
127126ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd𝑘) “ (1...𝑀)))
128125ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (1...𝑀))
129102, 127, 1283eqtr3d 2663 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) “ (1...𝑀)) = (1...𝑀))
130 f1oeq3 6086 . . . . . . . . . . . . . 14 (((2nd𝑘) “ (1...𝑀)) = (1...𝑀) → (((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd𝑘) “ (1...𝑀)) ↔ ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀)))
131129, 130syl 17 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→((2nd𝑘) “ (1...𝑀)) ↔ ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀)))
13281, 131mpbid 222 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀))
13373resex 5402 . . . . . . . . . . . . 13 ((2nd𝑘) ↾ (1...𝑀)) ∈ V
134 f1oeq1 6084 . . . . . . . . . . . . 13 (𝑓 = ((2nd𝑘) ↾ (1...𝑀)) → (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) ↔ ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀)))
135133, 134elab 3333 . . . . . . . . . . . 12 (((2nd𝑘) ↾ (1...𝑀)) ∈ {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ↔ ((2nd𝑘) ↾ (1...𝑀)):(1...𝑀)–1-1-onto→(1...𝑀))
136132, 135sylibr 224 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) ↾ (1...𝑀)) ∈ {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})
137 opelxpi 5108 . . . . . . . . . . 11 ((((1st𝑘) ↾ (1...𝑀)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) ∧ ((2nd𝑘) ↾ (1...𝑀)) ∈ {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))
13871, 136, 137syl2anc 692 . . . . . . . . . 10 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))
1391383ad2antr3 1226 . . . . . . . . 9 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))
140 3anass 1040 . . . . . . . . . . 11 ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))))
141 ancom 466 . . . . . . . . . . 11 ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ↔ ((((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
142140, 141bitri 264 . . . . . . . . . 10 ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ ((((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
14394nnzd 11425 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑀 + 1) ∈ ℤ)
144 uzid 11646 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑀 + 1) ∈ ℤ → (𝑀 + 1) ∈ (ℤ‘(𝑀 + 1)))
145 peano2uz 11685 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑀 + 1) ∈ (ℤ‘(𝑀 + 1)) → ((𝑀 + 1) + 1) ∈ (ℤ‘(𝑀 + 1)))
146143, 144, 1453syl 18 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((𝑀 + 1) + 1) ∈ (ℤ‘(𝑀 + 1)))
1475nn0zd 11424 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑀 ∈ ℤ)
1481nnzd 11425 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑁 ∈ ℤ)
149 zltp1le 11371 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁))
150 peano2z 11362 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ)
151 eluz 11645 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁))
152150, 151sylan 488 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘(𝑀 + 1)) ↔ (𝑀 + 1) ≤ 𝑁))
153149, 152bitr4d 271 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁𝑁 ∈ (ℤ‘(𝑀 + 1))))
154147, 148, 153syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑀 < 𝑁𝑁 ∈ (ℤ‘(𝑀 + 1))))
1557, 154mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 ∈ (ℤ‘(𝑀 + 1)))
156 fzsplit2 12308 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑀 + 1) + 1) ∈ (ℤ‘(𝑀 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
157146, 155, 156syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑀 + 1)...𝑁) = (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))
158 fzsn 12325 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑀 + 1) ∈ ℤ → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)})
159143, 158syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((𝑀 + 1)...(𝑀 + 1)) = {(𝑀 + 1)})
160159uneq1d 3744 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((𝑀 + 1)...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)))
161157, 160eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((𝑀 + 1)...𝑁) = ({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)))
162161xpeq1d 5098 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (((𝑀 + 1)...𝑁) × {0}) = (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}))
163162uneq2d 3745 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0})))
164 xpundir 5133 . . . . . . . . . . . . . . . . . . . . . . 23 (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
165 ovex 6632 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 + 1) ∈ V
166 c0ex 9978 . . . . . . . . . . . . . . . . . . . . . . . . 25 0 ∈ V
167165, 166xpsn 6361 . . . . . . . . . . . . . . . . . . . . . . . 24 ({(𝑀 + 1)} × {0}) = {⟨(𝑀 + 1), 0⟩}
168167uneq1i 3741 . . . . . . . . . . . . . . . . . . . . . . 23 (({(𝑀 + 1)} × {0}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
169164, 168eqtri 2643 . . . . . . . . . . . . . . . . . . . . . 22 (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0}) = ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
170169uneq2i 3742 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
171 unass 3748 . . . . . . . . . . . . . . . . . . . . 21 (((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ({⟨(𝑀 + 1), 0⟩} ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
172170, 171eqtr4i 2646 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (({(𝑀 + 1)} ∪ (((𝑀 + 1) + 1)...𝑁)) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0}))
173163, 172syl6eq 2671 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
174173ad3antrrr 765 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})) = (((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
175165a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 + 1) ∈ V)
176166a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → 0 ∈ V)
177110eqcomd 2627 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1)))
178177ad3antrrr 765 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((1...𝑀) ∪ {(𝑀 + 1)}) = (1...(𝑀 + 1)))
179 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = (𝑀 + 1) → ((1st𝑘)‘𝑛) = ((1st𝑘)‘(𝑀 + 1)))
180 fveq2 6148 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = (𝑀 + 1) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)))
181179, 180oveq12d 6622 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = (𝑀 + 1) → (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = (((1st𝑘)‘(𝑀 + 1)) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1))))
182 simplrl 799 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((1st𝑘)‘(𝑀 + 1)) = 0)
183 imain 5932 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Fun (2nd𝑘) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
18476, 83, 1833syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
185184ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
186 elfznn0 12374 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0)
187186nn0red 11296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
188187ltp1d 10898 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (0...𝑀) → 𝑗 < (𝑗 + 1))
189 fzdisj 12310 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅)
190188, 189syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1))) = ∅)
191190imaeq2d 5425 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑀) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ((2nd𝑘) “ ∅))
192 ima0 5440 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((2nd𝑘) “ ∅) = ∅
193191, 192syl6eq 2671 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑀) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
194185, 193sylan9req 2676 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
195 simplr 791 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))
19692ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (2nd𝑘) Fn (1...(𝑀 + 1)))
197 nn0p1nn 11276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ ℕ0 → (𝑗 + 1) ∈ ℕ)
198 nnuz 11667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ℕ = (ℤ‘1)
199197, 198syl6eleq 2708 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ ℕ0 → (𝑗 + 1) ∈ (ℤ‘1))
200 fzss1 12322 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑗 + 1) ∈ (ℤ‘1) → ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)))
201186, 199, 2003syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑀) → ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)))
202 elfzuz3 12281 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (0...𝑀) → 𝑀 ∈ (ℤ𝑗))
203 eluzp1p1 11657 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑀 ∈ (ℤ𝑗) → (𝑀 + 1) ∈ (ℤ‘(𝑗 + 1)))
204 eluzfz2 12291 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑀 + 1) ∈ (ℤ‘(𝑗 + 1)) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))
205202, 203, 2043syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))
206201, 205jca 554 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑀) → (((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1))))
207 fnfvima 6450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((2nd𝑘) Fn (1...(𝑀 + 1)) ∧ ((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1))) → ((2nd𝑘)‘(𝑀 + 1)) ∈ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
2082073expb 1263 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((2nd𝑘) Fn (1...(𝑀 + 1)) ∧ (((𝑗 + 1)...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ ((𝑗 + 1)...(𝑀 + 1)))) → ((2nd𝑘)‘(𝑀 + 1)) ∈ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
209196, 206, 208syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd𝑘)‘(𝑀 + 1)) ∈ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
210195, 209eqeltrrd 2699 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (𝑀 + 1) ∈ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
211 1ex 9979 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ V
212 fnconstg 6050 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (1 ∈ V → (((2nd𝑘) “ (1...𝑗)) × {1}) Fn ((2nd𝑘) “ (1...𝑗)))
213211, 212ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd𝑘) “ (1...𝑗)) × {1}) Fn ((2nd𝑘) “ (1...𝑗))
214 fnconstg 6050 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 ∈ V → (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
215166, 214ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))
216 fvun2 6227 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((2nd𝑘) “ (1...𝑗)) × {1}) Fn ((2nd𝑘) “ (1...𝑗)) ∧ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) ∧ ((((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅ ∧ (𝑀 + 1) ∈ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
217213, 215, 216mp3an12 1411 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅ ∧ (𝑀 + 1) ∈ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
218194, 210, 217syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = ((((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)))
219166fvconst2 6423 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑀 + 1) ∈ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) → ((((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
220210, 219syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})‘(𝑀 + 1)) = 0)
221218, 220eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = 0)
222221adantlrl 755 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1)) = 0)
223182, 222oveq12d 6622 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st𝑘)‘(𝑀 + 1)) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1))) = (0 + 0))
224 00id 10155 . . . . . . . . . . . . . . . . . . . . . 22 (0 + 0) = 0
225223, 224syl6eq 2671 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st𝑘)‘(𝑀 + 1)) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘(𝑀 + 1))) = 0)
226181, 225sylan9eqr 2677 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 = (𝑀 + 1)) → (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛)) = 0)
227175, 176, 178, 226fmptapd 6391 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) = (𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
228227uneq1d 3744 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ {⟨(𝑀 + 1), 0⟩}) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
229174, 228eqtrd 2655 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
230 elmapfn 7824 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑘) ∈ ((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) → (1st𝑘) Fn (1...(𝑀 + 1)))
23161, 230syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → (1st𝑘) Fn (1...(𝑀 + 1)))
232 fnssres 5962 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑘) Fn (1...(𝑀 + 1)) ∧ (1...𝑀) ⊆ (1...(𝑀 + 1))) → ((1st𝑘) ↾ (1...𝑀)) Fn (1...𝑀))
233231, 64, 232sylancl 693 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((1st𝑘) ↾ (1...𝑀)) Fn (1...𝑀))
234233ad3antlr 766 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1st𝑘) ↾ (1...𝑀)) Fn (1...𝑀))
235 simplr 791 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}))
236 fnconstg 6050 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0 ∈ V → (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...𝑀)))
237166, 236ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...𝑀))
238213, 237pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((2nd𝑘) “ (1...𝑗)) × {1}) Fn ((2nd𝑘) “ (1...𝑗)) ∧ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...𝑀)))
239 imain 5932 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Fun (2nd𝑘) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))))
24076, 83, 2393syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))))
241 fzdisj 12310 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 < (𝑗 + 1) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅)
242188, 241syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑀) → ((1...𝑗) ∩ ((𝑗 + 1)...𝑀)) = ∅)
243242imaeq2d 5425 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑀) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ((2nd𝑘) “ ∅))
244243, 192syl6eq 2671 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑀) → ((2nd𝑘) “ ((1...𝑗) ∩ ((𝑗 + 1)...𝑀))) = ∅)
245240, 244sylan9req 2676 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))) = ∅)
246 fnun 5955 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((2nd𝑘) “ (1...𝑗)) × {1}) Fn ((2nd𝑘) “ (1...𝑗)) ∧ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...𝑀))) ∧ (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))) = ∅) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))))
247238, 245, 246sylancr 694 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))))
248235, 247sylan 488 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))))
249101adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd𝑘) “ {(𝑀 + 1)})) = ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}))
25085ad3antlr 766 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = (((2nd𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd𝑘) “ {(𝑀 + 1)})))
251186, 197syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ ℕ)
252251, 198syl6eleq 2708 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ (ℤ‘1))
253 fzsplit2 12308 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑗 + 1) ∈ (ℤ‘1) ∧ 𝑀 ∈ (ℤ𝑗)) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
254252, 202, 253syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑀) → (1...𝑀) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
255128, 254sylan9eq 2675 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = ((1...𝑗) ∪ ((𝑗 + 1)...𝑀)))
256255imaeq2d 5425 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((2nd𝑘) “ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))))
257250, 256eqtr3d 2657 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd𝑘) “ (1...(𝑀 + 1))) ∖ ((2nd𝑘) “ {(𝑀 + 1)})) = ((2nd𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))))
258125ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)}) = (1...𝑀))
259249, 257, 2583eqtr3rd 2664 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (1...𝑀) = ((2nd𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))))
260 imaundi 5504 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2nd𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...𝑀))) = (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...𝑀)))
261259, 260syl6eq 2671 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (1...𝑀) = (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...𝑀))))
262261fneq2d 5940 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ↔ ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...𝑀)))))
263248, 262mpbird 247 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀))
264 fzss2 12323 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑀 ∈ (ℤ𝑗) → (1...𝑗) ⊆ (1...𝑀))
265 resima2 5391 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((1...𝑗) ⊆ (1...𝑀) → (((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) = ((2nd𝑘) “ (1...𝑗)))
266202, 264, 2653syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑀) → (((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) = ((2nd𝑘) “ (1...𝑗)))
267266xpeq1d 5098 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0...𝑀) → ((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) = (((2nd𝑘) “ (1...𝑗)) × {1}))
268186, 199syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ (ℤ‘1))
269 fzss1 12322 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑗 + 1) ∈ (ℤ‘1) → ((𝑗 + 1)...𝑀) ⊆ (1...𝑀))
270 resima2 5391 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑗 + 1)...𝑀) ⊆ (1...𝑀) → (((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) = ((2nd𝑘) “ ((𝑗 + 1)...𝑀)))
271268, 269, 2703syl 18 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑀) → (((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) = ((2nd𝑘) “ ((𝑗 + 1)...𝑀)))
272271xpeq1d 5098 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0...𝑀) → ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}) = (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))
273267, 272uneq12d 3746 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (0...𝑀) → (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) = ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})))
274273adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) = ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})))
275274fneq1d 5939 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ↔ ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀)))
276263, 275mpbird 247 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀))
277 fzfid 12712 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (1...𝑀) ∈ Fin)
278 inidm 3800 . . . . . . . . . . . . . . . . . . . 20 ((1...𝑀) ∩ (1...𝑀)) = (1...𝑀)
279 fvres 6164 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ (1...𝑀) → (((1st𝑘) ↾ (1...𝑀))‘𝑛) = ((1st𝑘)‘𝑛))
280279adantl 482 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (((1st𝑘) ↾ (1...𝑀))‘𝑛) = ((1st𝑘)‘𝑛))
281 disjsn 4216 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ↔ ¬ (𝑀 + 1) ∈ (1...𝑀))
282122, 281sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)
283282ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅)
284263, 283jca 554 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅))
285 fnconstg 6050 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ∈ V → ({(𝑀 + 1)} × {0}) Fn {(𝑀 + 1)})
286166, 285ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . 24 ({(𝑀 + 1)} × {0}) Fn {(𝑀 + 1)}
287 fvun1 6226 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ ({(𝑀 + 1)} × {0}) Fn {(𝑀 + 1)} ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ 𝑛 ∈ (1...𝑀))) → ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
288286, 287mp3an2 1409 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ (((1...𝑀) ∩ {(𝑀 + 1)}) = ∅ ∧ 𝑛 ∈ (1...𝑀))) → ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
289288anassrs 679 . . . . . . . . . . . . . . . . . . . . . 22 (((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) Fn (1...𝑀) ∧ ((1...𝑀) ∩ {(𝑀 + 1)}) = ∅) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
290284, 289sylan 488 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
291251nnzd 11425 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑗 ∈ (0...𝑀) → (𝑗 + 1) ∈ ℤ)
292186nn0cnd 11297 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℂ)
293 pncan1 10398 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑗 ∈ ℂ → ((𝑗 + 1) − 1) = 𝑗)
294292, 293syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑗 ∈ (0...𝑀) → ((𝑗 + 1) − 1) = 𝑗)
295294fveq2d 6152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑗 ∈ (0...𝑀) → (ℤ‘((𝑗 + 1) − 1)) = (ℤ𝑗))
296202, 295eleqtrrd 2701 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑗 ∈ (0...𝑀) → 𝑀 ∈ (ℤ‘((𝑗 + 1) − 1)))
297 fzsuc2 12340 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑗 + 1) ∈ ℤ ∧ 𝑀 ∈ (ℤ‘((𝑗 + 1) − 1))) → ((𝑗 + 1)...(𝑀 + 1)) = (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)}))
298291, 296, 297syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑗 ∈ (0...𝑀) → ((𝑗 + 1)...(𝑀 + 1)) = (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)}))
299298imaeq2d 5425 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 ∈ (0...𝑀) → ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) = ((2nd𝑘) “ (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)})))
300 imaundi 5504 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((2nd𝑘) “ (((𝑗 + 1)...𝑀) ∪ {(𝑀 + 1)})) = (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd𝑘) “ {(𝑀 + 1)}))
301299, 300syl6eq 2671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ (0...𝑀) → ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) = (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd𝑘) “ {(𝑀 + 1)})))
302301xpeq1d 5098 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (0...𝑀) → (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd𝑘) “ {(𝑀 + 1)})) × {0}))
303 xpundir 5133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((2nd𝑘) “ ((𝑗 + 1)...𝑀)) ∪ ((2nd𝑘) “ {(𝑀 + 1)})) × {0}) = ((((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0}))
304302, 303syl6eq 2671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (0...𝑀) → (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = ((((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0})))
305304uneq2d 3745 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (0...𝑀) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0}))))
306 unass 3748 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0})) = ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0})))
307305, 306syl6eqr 2673 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (0...𝑀) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0})))
308307adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0})))
30998xpeq1d 5098 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ({((2nd𝑘)‘(𝑀 + 1))} × {0}) = (((2nd𝑘) “ {(𝑀 + 1)}) × {0}))
310309uneq2d 3745 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({((2nd𝑘)‘(𝑀 + 1))} × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0})))
311310adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({((2nd𝑘)‘(𝑀 + 1))} × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ (((2nd𝑘) “ {(𝑀 + 1)}) × {0})))
312308, 311eqtr4d 2658 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({((2nd𝑘)‘(𝑀 + 1))} × {0})))
31399xpeq1d 5098 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → ({((2nd𝑘)‘(𝑀 + 1))} × {0}) = ({(𝑀 + 1)} × {0}))
314313uneq2d 3745 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({((2nd𝑘)‘(𝑀 + 1))} × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0})))
315312, 314sylan9eq 2675 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ 𝑗 ∈ (0...𝑀)) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0})))
316315an32s 845 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0})))
317316fveq1d 6150 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛))
318317adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = ((((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0})) ∪ ({(𝑀 + 1)} × {0}))‘𝑛))
319273fveq1d 6150 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (0...𝑀) → ((((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
320319ad2antlr 762 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛))
321290, 318, 3203eqtr4rd 2666 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...𝑀)) → ((((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))
322234, 276, 277, 277, 278, 280, 321offval 6857 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → (((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) = (𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
323322uneq1d 3744 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ 𝑗 ∈ (0...𝑀)) → ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})))
324323adantlrl 755 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...𝑀) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ (((𝑀 + 1)...𝑁) × {0})))
325 simplr 791 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}))
326231adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (1st𝑘) Fn (1...(𝑀 + 1)))
327213, 215pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . 22 ((((2nd𝑘) “ (1...𝑗)) × {1}) Fn ((2nd𝑘) “ (1...𝑗)) ∧ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
328184, 193sylan9req 2676 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅)
329 fnun 5955 . . . . . . . . . . . . . . . . . . . . . 22 ((((((2nd𝑘) “ (1...𝑗)) × {1}) Fn ((2nd𝑘) “ (1...𝑗)) ∧ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) Fn ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) ∧ (((2nd𝑘) “ (1...𝑗)) ∩ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ∅) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
330327, 328, 329sylancr 694 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))))
331 peano2uz 11685 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑀 ∈ (ℤ𝑗) → (𝑀 + 1) ∈ (ℤ𝑗))
332202, 331syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ (0...𝑀) → (𝑀 + 1) ∈ (ℤ𝑗))
333 fzsplit2 12308 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑗 + 1) ∈ (ℤ‘1) ∧ (𝑀 + 1) ∈ (ℤ𝑗)) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))
334268, 332, 333syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0...𝑀) → (1...(𝑀 + 1)) = ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1))))
335334imaeq2d 5425 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0...𝑀) → ((2nd𝑘) “ (1...(𝑀 + 1))) = ((2nd𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))))
336 imaundi 5504 . . . . . . . . . . . . . . . . . . . . . . . 24 ((2nd𝑘) “ ((1...𝑗) ∪ ((𝑗 + 1)...(𝑀 + 1)))) = (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
337335, 336syl6req 2672 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (0...𝑀) → (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = ((2nd𝑘) “ (1...(𝑀 + 1))))
338337, 89sylan9eqr 2677 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) = (1...(𝑀 + 1)))
339338fneq2d 5940 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (((2nd𝑘) “ (1...𝑗)) ∪ ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1)))) ↔ ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (1...(𝑀 + 1))))
340330, 339mpbid 222 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) Fn (1...(𝑀 + 1)))
341 fzfid 12712 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (1...(𝑀 + 1)) ∈ Fin)
342 inidm 3800 . . . . . . . . . . . . . . . . . . . 20 ((1...(𝑀 + 1)) ∩ (1...(𝑀 + 1))) = (1...(𝑀 + 1))
343 eqidd 2622 . . . . . . . . . . . . . . . . . . . 20 (((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...(𝑀 + 1))) → ((1st𝑘)‘𝑛) = ((1st𝑘)‘𝑛))
344 eqidd 2622 . . . . . . . . . . . . . . . . . . . 20 (((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑛 ∈ (1...(𝑀 + 1))) → (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛) = (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))
345326, 340, 341, 341, 342, 343, 344offval 6857 . . . . . . . . . . . . . . . . . . 19 ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → ((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = (𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))))
346345uneq1d 3744 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∧ 𝑗 ∈ (0...𝑀)) → (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
347325, 346sylan 488 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((𝑛 ∈ (1...(𝑀 + 1)) ↦ (((1st𝑘)‘𝑛) + (((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))‘𝑛))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
348229, 324, 3473eqtr4rd 2666 . . . . . . . . . . . . . . . 16 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})))
349348csbeq1d 3521 . . . . . . . . . . . . . . 15 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵)
350349eqeq2d 2631 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) ∧ 𝑗 ∈ (0...𝑀)) → (𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
351350rexbidva 3042 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
352351ralbidv 2980 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
353352biimpd 219 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
354353impr 648 . . . . . . . . . 10 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) ∧ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵)) → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵)
355142, 354sylan2b 492 . . . . . . . . 9 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵)
356 1st2nd2 7150 . . . . . . . . . . . 12 (𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) → 𝑘 = ⟨(1st𝑘), (2nd𝑘)⟩)
357356ad2antlr 762 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 = ⟨(1st𝑘), (2nd𝑘)⟩)
358 fnsnsplit 6404 . . . . . . . . . . . . . . . 16 (((1st𝑘) Fn (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (1...(𝑀 + 1))) → (1st𝑘) = (((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩}))
359231, 96, 358syl2anr 495 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → (1st𝑘) = (((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩}))
360359adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((1st𝑘)‘(𝑀 + 1)) = 0) → (1st𝑘) = (((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩}))
361125reseq2d 5356 . . . . . . . . . . . . . . . 16 (𝜑 → ((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1st𝑘) ↾ (1...𝑀)))
362361adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1st𝑘) ↾ (1...𝑀)))
363 opeq2 4371 . . . . . . . . . . . . . . . 16 (((1st𝑘)‘(𝑀 + 1)) = 0 → ⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩ = ⟨(𝑀 + 1), 0⟩)
364363sneqd 4160 . . . . . . . . . . . . . . 15 (((1st𝑘)‘(𝑀 + 1)) = 0 → {⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩} = {⟨(𝑀 + 1), 0⟩})
365 uneq12 3740 . . . . . . . . . . . . . . 15 ((((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((1st𝑘) ↾ (1...𝑀)) ∧ {⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩} = {⟨(𝑀 + 1), 0⟩}) → (((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩}) = (((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
366362, 364, 365syl2an 494 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((1st𝑘)‘(𝑀 + 1)) = 0) → (((1st𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((1st𝑘)‘(𝑀 + 1))⟩}) = (((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
367360, 366eqtrd 2655 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((1st𝑘)‘(𝑀 + 1)) = 0) → (1st𝑘) = (((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
368367adantrr 752 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (1st𝑘) = (((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
369 fnsnsplit 6404 . . . . . . . . . . . . . . . 16 (((2nd𝑘) Fn (1...(𝑀 + 1)) ∧ (𝑀 + 1) ∈ (1...(𝑀 + 1))) → (2nd𝑘) = (((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩}))
37092, 96, 369syl2anr 495 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → (2nd𝑘) = (((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩}))
371370adantr 481 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (2nd𝑘) = (((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩}))
372125reseq2d 5356 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd𝑘) ↾ (1...𝑀)))
373372adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd𝑘) ↾ (1...𝑀)))
374 opeq2 4371 . . . . . . . . . . . . . . . 16 (((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → ⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩ = ⟨(𝑀 + 1), (𝑀 + 1)⟩)
375374sneqd 4160 . . . . . . . . . . . . . . 15 (((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1) → {⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩} = {⟨(𝑀 + 1), (𝑀 + 1)⟩})
376 uneq12 3740 . . . . . . . . . . . . . . 15 ((((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) = ((2nd𝑘) ↾ (1...𝑀)) ∧ {⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩} = {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → (((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩}) = (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
377373, 375, 376syl2an 494 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (((2nd𝑘) ↾ ((1...(𝑀 + 1)) ∖ {(𝑀 + 1)})) ∪ {⟨(𝑀 + 1), ((2nd𝑘)‘(𝑀 + 1))⟩}) = (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
378371, 377eqtrd 2655 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (2nd𝑘) = (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
379378adantrl 751 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → (2nd𝑘) = (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
380368, 379opeq12d 4378 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → ⟨(1st𝑘), (2nd𝑘)⟩ = ⟨(((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
381357, 380eqtrd 2655 . . . . . . . . . 10 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 = ⟨(((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
3823813adantr1 1218 . . . . . . . . 9 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → 𝑘 = ⟨(((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
383 fvex 6158 . . . . . . . . . . . . . . . . . . 19 (1st𝑘) ∈ V
384383resex 5402 . . . . . . . . . . . . . . . . . 18 ((1st𝑘) ↾ (1...𝑀)) ∈ V
385384, 133op1std 7123 . . . . . . . . . . . . . . . . 17 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (1st𝑡) = ((1st𝑘) ↾ (1...𝑀)))
386384, 133op2ndd 7124 . . . . . . . . . . . . . . . . . . . 20 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (2nd𝑡) = ((2nd𝑘) ↾ (1...𝑀)))
387386imaeq1d 5424 . . . . . . . . . . . . . . . . . . 19 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ((2nd𝑡) “ (1...𝑗)) = (((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)))
388387xpeq1d 5098 . . . . . . . . . . . . . . . . . 18 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (((2nd𝑡) “ (1...𝑗)) × {1}) = ((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}))
389386imaeq1d 5424 . . . . . . . . . . . . . . . . . . 19 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ((2nd𝑡) “ ((𝑗 + 1)...𝑀)) = (((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)))
390389xpeq1d 5098 . . . . . . . . . . . . . . . . . 18 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}) = ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))
391388, 390uneq12d 3746 . . . . . . . . . . . . . . . . 17 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0})) = (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0})))
392385, 391oveq12d 6622 . . . . . . . . . . . . . . . 16 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) = (((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))))
393392uneq1d 3744 . . . . . . . . . . . . . . 15 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})))
394393csbeq1d 3521 . . . . . . . . . . . . . 14 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵)
395394eqeq2d 2631 . . . . . . . . . . . . 13 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
396395rexbidv 3045 . . . . . . . . . . . 12 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
397396ralbidv 2980 . . . . . . . . . . 11 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵))
398385uneq1d 3744 . . . . . . . . . . . . 13 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = (((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}))
399386uneq1d 3744 . . . . . . . . . . . . 13 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
400398, 399opeq12d 4378 . . . . . . . . . . . 12 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨(((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
401400eqeq2d 2631 . . . . . . . . . . 11 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → (𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ 𝑘 = ⟨(((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
402397, 401anbi12d 746 . . . . . . . . . 10 (𝑡 = ⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨(((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)))
403402rspcev 3295 . . . . . . . . 9 ((⟨((1st𝑘) ↾ (1...𝑀)), ((2nd𝑘) ↾ (1...𝑀))⟩ ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = ((((1st𝑘) ↾ (1...𝑀)) ∘𝑓 + (((((2nd𝑘) ↾ (1...𝑀)) “ (1...𝑗)) × {1}) ∪ ((((2nd𝑘) ↾ (1...𝑀)) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨(((1st𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), 0⟩}), (((2nd𝑘) ↾ (1...𝑀)) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)) → ∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
404139, 355, 382, 403syl12anc 1321 . . . . . . . 8 (((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) ∧ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))) → ∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
405404ex 450 . . . . . . 7 ((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)))
406 elrabi 3342 . . . . . . . . . . 11 (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} → 𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))
407 elrabi 3342 . . . . . . . . . . 11 (𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} → 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))
408406, 407anim12i 589 . . . . . . . . . 10 ((𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ∧ 𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}) → (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})))
409 eqtr2 2641 . . . . . . . . . . . 12 ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
41022, 24opth 4905 . . . . . . . . . . . . 13 (⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
411 difeq1 3699 . . . . . . . . . . . . . . 15 (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) → (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∖ {⟨(𝑀 + 1), 0⟩}) = (((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) ∖ {⟨(𝑀 + 1), 0⟩}))
412 difun2 4020 . . . . . . . . . . . . . . 15 (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩})
413 difun2 4020 . . . . . . . . . . . . . . 15 (((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩})
414411, 412, 4133eqtr3g 2678 . . . . . . . . . . . . . 14 (((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) → ((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}))
415 difeq1 3699 . . . . . . . . . . . . . . 15 (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
416 difun2 4020 . . . . . . . . . . . . . . 15 (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})
417 difun2 4020 . . . . . . . . . . . . . . 15 (((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})
418415, 416, 4173eqtr3g 2678 . . . . . . . . . . . . . 14 (((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) → ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
419414, 418anim12i 589 . . . . . . . . . . . . 13 ((((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})) → (((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
420410, 419sylbi 207 . . . . . . . . . . . 12 (⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ → (((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
421409, 420syl 17 . . . . . . . . . . 11 ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → (((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})))
422 elmapfn 7824 . . . . . . . . . . . . . . . . . . 19 ((1st𝑡) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) → (1st𝑡) Fn (1...𝑀))
423 fnop 5952 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑡) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑡)) → (𝑀 + 1) ∈ (1...𝑀))
424423ex 450 . . . . . . . . . . . . . . . . . . 19 ((1st𝑡) Fn (1...𝑀) → (⟨(𝑀 + 1), 0⟩ ∈ (1st𝑡) → (𝑀 + 1) ∈ (1...𝑀)))
4259, 422, 4243syl 18 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (⟨(𝑀 + 1), 0⟩ ∈ (1st𝑡) → (𝑀 + 1) ∈ (1...𝑀)))
426425, 122nsyli 155 . . . . . . . . . . . . . . . . 17 (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑡)))
427426impcom 446 . . . . . . . . . . . . . . . 16 ((𝜑𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑡))
428 difsn 4297 . . . . . . . . . . . . . . . 16 (¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑡) → ((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = (1st𝑡))
429427, 428syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = (1st𝑡))
430 xp1st 7143 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (1st𝑛) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)))
431 elmapfn 7824 . . . . . . . . . . . . . . . . . . 19 ((1st𝑛) ∈ ((0..^𝐾) ↑𝑚 (1...𝑀)) → (1st𝑛) Fn (1...𝑀))
432 fnop 5952 . . . . . . . . . . . . . . . . . . . 20 (((1st𝑛) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑛)) → (𝑀 + 1) ∈ (1...𝑀))
433432ex 450 . . . . . . . . . . . . . . . . . . 19 ((1st𝑛) Fn (1...𝑀) → (⟨(𝑀 + 1), 0⟩ ∈ (1st𝑛) → (𝑀 + 1) ∈ (1...𝑀)))
434430, 431, 4333syl 18 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (⟨(𝑀 + 1), 0⟩ ∈ (1st𝑛) → (𝑀 + 1) ∈ (1...𝑀)))
435434, 122nsyli 155 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑛)))
436435impcom 446 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑛))
437 difsn 4297 . . . . . . . . . . . . . . . 16 (¬ ⟨(𝑀 + 1), 0⟩ ∈ (1st𝑛) → ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) = (1st𝑛))
438436, 437syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) = (1st𝑛))
439429, 438eqeqan12d 2637 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) ∧ (𝜑𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ↔ (1st𝑡) = (1st𝑛)))
440439anandis 872 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ↔ (1st𝑡) = (1st𝑛)))
441 f1ofn 6095 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑡):(1...𝑀)–1-1-onto→(1...𝑀) → (2nd𝑡) Fn (1...𝑀))
442 fnop 5952 . . . . . . . . . . . . . . . . . . . 20 (((2nd𝑡) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑡)) → (𝑀 + 1) ∈ (1...𝑀))
443442ex 450 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑡) Fn (1...𝑀) → (⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑡) → (𝑀 + 1) ∈ (1...𝑀)))
44417, 441, 4433syl 18 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑡) → (𝑀 + 1) ∈ (1...𝑀)))
445444, 122nsyli 155 . . . . . . . . . . . . . . . . 17 (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑡)))
446445impcom 446 . . . . . . . . . . . . . . . 16 ((𝜑𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑡))
447 difsn 4297 . . . . . . . . . . . . . . . 16 (¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑡) → ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (2nd𝑡))
448446, 447syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (2nd𝑡))
449 xp2nd 7144 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd𝑛) ∈ {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})
450 fvex 6158 . . . . . . . . . . . . . . . . . . . . 21 (2nd𝑛) ∈ V
451 f1oeq1 6084 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 = (2nd𝑛) → (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) ↔ (2nd𝑛):(1...𝑀)–1-1-onto→(1...𝑀)))
452450, 451elab 3333 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑛) ∈ {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ↔ (2nd𝑛):(1...𝑀)–1-1-onto→(1...𝑀))
453449, 452sylib 208 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (2nd𝑛):(1...𝑀)–1-1-onto→(1...𝑀))
454 f1ofn 6095 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑛):(1...𝑀)–1-1-onto→(1...𝑀) → (2nd𝑛) Fn (1...𝑀))
455 fnop 5952 . . . . . . . . . . . . . . . . . . . 20 (((2nd𝑛) Fn (1...𝑀) ∧ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑛)) → (𝑀 + 1) ∈ (1...𝑀))
456455ex 450 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑛) Fn (1...𝑀) → (⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑛) → (𝑀 + 1) ∈ (1...𝑀)))
457453, 454, 4563syl 18 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑛) → (𝑀 + 1) ∈ (1...𝑀)))
458457, 122nsyli 155 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) → (𝜑 → ¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑛)))
459458impcom 446 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑛))
460 difsn 4297 . . . . . . . . . . . . . . . 16 (¬ ⟨(𝑀 + 1), (𝑀 + 1)⟩ ∈ (2nd𝑛) → ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (2nd𝑛))
461459, 460syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = (2nd𝑛))
462448, 461eqeqan12d 2637 . . . . . . . . . . . . . 14 (((𝜑𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) ∧ (𝜑𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ↔ (2nd𝑡) = (2nd𝑛)))
463462anandis 872 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) ↔ (2nd𝑡) = (2nd𝑛)))
464440, 463anbi12d 746 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → ((((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})) ↔ ((1st𝑡) = (1st𝑛) ∧ (2nd𝑡) = (2nd𝑛))))
465 xpopth 7152 . . . . . . . . . . . . 13 ((𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})) → (((1st𝑡) = (1st𝑛) ∧ (2nd𝑡) = (2nd𝑛)) ↔ 𝑡 = 𝑛))
466465adantl 482 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → (((1st𝑡) = (1st𝑛) ∧ (2nd𝑡) = (2nd𝑛)) ↔ 𝑡 = 𝑛))
467464, 466bitrd 268 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → ((((1st𝑡) ∖ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∖ {⟨(𝑀 + 1), 0⟩}) ∧ ((2nd𝑡) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∖ {⟨(𝑀 + 1), (𝑀 + 1)⟩})) ↔ 𝑡 = 𝑛))
468421, 467syl5ib 234 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∧ 𝑛 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}))) → ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))
469408, 468sylan2 491 . . . . . . . . 9 ((𝜑 ∧ (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ∧ 𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵})) → ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))
470469ralrimivva 2965 . . . . . . . 8 (𝜑 → ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))
471470adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))
472405, 471jctird 566 . . . . . 6 ((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → (∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛))))
473 fveq2 6148 . . . . . . . . . . 11 (𝑡 = 𝑛 → (1st𝑡) = (1st𝑛))
474473uneq1d 3744 . . . . . . . . . 10 (𝑡 = 𝑛 → ((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}) = ((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}))
475 fveq2 6148 . . . . . . . . . . 11 (𝑡 = 𝑛 → (2nd𝑡) = (2nd𝑛))
476475uneq1d 3744 . . . . . . . . . 10 (𝑡 = 𝑛 → ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}) = ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩}))
477474, 476opeq12d 4378 . . . . . . . . 9 (𝑡 = 𝑛 → ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
478477eqeq2d 2631 . . . . . . . 8 (𝑡 = 𝑛 → (𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
479478reu4 3382 . . . . . . 7 (∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ (∃𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛)))
48058rexrab 3352 . . . . . . . 8 (∃𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ ∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
481480anbi1i 730 . . . . . . 7 ((∃𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛)) ↔ (∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛)))
482479, 481bitri 264 . . . . . 6 (∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ (∃𝑡 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)})(∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑡) ∘𝑓 + ((((2nd𝑡) “ (1...𝑗)) × {1}) ∪ (((2nd𝑡) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) ∧ ∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}∀𝑛 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ((𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∧ 𝑘 = ⟨((1st𝑛) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑛) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) → 𝑡 = 𝑛)))
483472, 482syl6ibr 242 . . . . 5 ((𝜑𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})) → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
484483ralrimiva 2960 . . . 4 (𝜑 → ∀𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
485 fveq2 6148 . . . . . . . . . . . 12 (𝑠 = 𝑘 → (1st𝑠) = (1st𝑘))
486 fveq2 6148 . . . . . . . . . . . . . . 15 (𝑠 = 𝑘 → (2nd𝑠) = (2nd𝑘))
487486imaeq1d 5424 . . . . . . . . . . . . . 14 (𝑠 = 𝑘 → ((2nd𝑠) “ (1...𝑗)) = ((2nd𝑘) “ (1...𝑗)))
488487xpeq1d 5098 . . . . . . . . . . . . 13 (𝑠 = 𝑘 → (((2nd𝑠) “ (1...𝑗)) × {1}) = (((2nd𝑘) “ (1...𝑗)) × {1}))
489486imaeq1d 5424 . . . . . . . . . . . . . 14 (𝑠 = 𝑘 → ((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) = ((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))))
490489xpeq1d 5098 . . . . . . . . . . . . 13 (𝑠 = 𝑘 → (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}) = (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))
491488, 490uneq12d 3746 . . . . . . . . . . . 12 (𝑠 = 𝑘 → ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})) = ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0})))
492485, 491oveq12d 6622 . . . . . . . . . . 11 (𝑠 = 𝑘 → ((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) = ((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))))
493492uneq1d 3744 . . . . . . . . . 10 (𝑠 = 𝑘 → (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})))
494493csbeq1d 3521 . . . . . . . . 9 (𝑠 = 𝑘(((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵)
495494eqeq2d 2631 . . . . . . . 8 (𝑠 = 𝑘 → (𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
496495rexbidv 3045 . . . . . . 7 (𝑠 = 𝑘 → (∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
497496ralbidv 2980 . . . . . 6 (𝑠 = 𝑘 → (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ↔ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵))
498485fveq1d 6150 . . . . . . 7 (𝑠 = 𝑘 → ((1st𝑠)‘(𝑀 + 1)) = ((1st𝑘)‘(𝑀 + 1)))
499498eqeq1d 2623 . . . . . 6 (𝑠 = 𝑘 → (((1st𝑠)‘(𝑀 + 1)) = 0 ↔ ((1st𝑘)‘(𝑀 + 1)) = 0))
500486fveq1d 6150 . . . . . . 7 (𝑠 = 𝑘 → ((2nd𝑠)‘(𝑀 + 1)) = ((2nd𝑘)‘(𝑀 + 1)))
501500eqeq1d 2623 . . . . . 6 (𝑠 = 𝑘 → (((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1) ↔ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)))
502497, 499, 5013anbi123d 1396 . . . . 5 (𝑠 = 𝑘 → ((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1)) ↔ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1))))
503502ralrab 3350 . . . 4 (∀𝑘 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ↔ ∀𝑘 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))})((∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑘) ∘𝑓 + ((((2nd𝑘) “ (1...𝑗)) × {1}) ∪ (((2nd𝑘) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑘)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑘)‘(𝑀 + 1)) = (𝑀 + 1)) → ∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
504484, 503sylibr 224 . . 3 (𝜑 → ∀𝑘 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
505 eqid 2621 . . . 4 (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ↦ ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩) = (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ↦ ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩)
506505f1ompt 6338 . . 3 ((𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ↦ ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩):{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ↔ (∀𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩ ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} ∧ ∀𝑘 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))}∃!𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}𝑘 = ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩))
50760, 504, 506sylanbrc 697 . 2 (𝜑 → (𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ↦ ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩):{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
508 ovex 6632 . . . . 5 ((0..^𝐾) ↑𝑚 (1...𝑀)) ∈ V
509 ovex 6632 . . . . . 6 ((1...𝑀) ↑𝑚 (1...𝑀)) ∈ V
510 f1of 6094 . . . . . . . 8 (𝑓:(1...𝑀)–1-1-onto→(1...𝑀) → 𝑓:(1...𝑀)⟶(1...𝑀))
511510ss2abi 3653 . . . . . . 7 {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ⊆ {𝑓𝑓:(1...𝑀)⟶(1...𝑀)}
51268, 68mapval 7814 . . . . . . 7 ((1...𝑀) ↑𝑚 (1...𝑀)) = {𝑓𝑓:(1...𝑀)⟶(1...𝑀)}
513511, 512sseqtr4i 3617 . . . . . 6 {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ⊆ ((1...𝑀) ↑𝑚 (1...𝑀))
514509, 513ssexi 4763 . . . . 5 {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)} ∈ V
515508, 514xpex 6915 . . . 4 (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∈ V
516515rabex 4773 . . 3 {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ∈ V
517516f1oen 7920 . 2 ((𝑡 ∈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ↦ ⟨((1st𝑡) ∪ {⟨(𝑀 + 1), 0⟩}), ((2nd𝑡) ∪ {⟨(𝑀 + 1), (𝑀 + 1)⟩})⟩):{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵}–1-1-onto→{𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))} → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
518507, 517syl 17 1 (𝜑 → {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...𝑀)) × {𝑓𝑓:(1...𝑀)–1-1-onto→(1...𝑀)}) ∣ ∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...𝑀)) × {0}))) ∪ (((𝑀 + 1)...𝑁) × {0})) / 𝑝𝐵} ≈ {𝑠 ∈ (((0..^𝐾) ↑𝑚 (1...(𝑀 + 1))) × {𝑓𝑓:(1...(𝑀 + 1))–1-1-onto→(1...(𝑀 + 1))}) ∣ (∀𝑖 ∈ (0...𝑀)∃𝑗 ∈ (0...𝑀)𝑖 = (((1st𝑠) ∘𝑓 + ((((2nd𝑠) “ (1...𝑗)) × {1}) ∪ (((2nd𝑠) “ ((𝑗 + 1)...(𝑀 + 1))) × {0}))) ∪ ((((𝑀 + 1) + 1)...𝑁) × {0})) / 𝑝𝐵 ∧ ((1st𝑠)‘(𝑀 + 1)) = 0 ∧ ((2nd𝑠)‘(𝑀 + 1)) = (𝑀 + 1))})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  {cab 2607  ∀wral 2907  ∃wrex 2908  ∃!wreu 2909  {crab 2911  Vcvv 3186  ⦋csb 3514   ∖ cdif 3552   ∪ cun 3553   ∩ cin 3554   ⊆ wss 3555  ∅c0 3891  {csn 4148  ⟨cop 4154   class class class wbr 4613   ↦ cmpt 4673   × cxp 5072  ◡ccnv 5073   ↾ cres 5076   “ cima 5077  Fun wfun 5841   Fn wfn 5842  ⟶wf 5843  –1-1→wf1 5844  –onto→wfo 5845  –1-1-onto→wf1o 5846  ‘cfv 5847  (class class class)co 6604   ∘𝑓 cof 6848  1st c1st 7111  2nd c2nd 7112   ↑𝑚 cmap 7802   ≈ cen 7896  Fincfn 7899  ℂcc 9878  ℝcr 9879  0cc0 9880  1c1 9881   + caddc 9883   < clt 10018   ≤ cle 10019   − cmin 10210  ℕcn 10964  ℕ0cn0 11236  ℤcz 11321  ℤ≥cuz 11631  ...cfz 12268  ..^cfzo 12406 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407 This theorem is referenced by:  poimirlem28  33069
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