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Theorem tfr1onlemaccex 6043
Description: We can define an acceptable function on any element of 𝑋.

As with many of the transfinite recursion theorems, we have hypotheses that state that 𝐹 is a function and that it is defined up to 𝑋. (Contributed by Jim Kingdon, 16-Mar-2022.)

Hypotheses
Ref Expression
tfr1on.f 𝐹 = recs(𝐺)
tfr1on.g (𝜑 → Fun 𝐺)
tfr1on.x (𝜑 → Ord 𝑋)
tfr1on.ex ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
tfr1onlemsucfn.1 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
tfr1onlemaccex.u ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
Assertion
Ref Expression
tfr1onlemaccex ((𝜑𝐶𝑋) → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
Distinct variable groups:   𝑢,𝐴,𝑥   𝐶,𝑔,𝑢   𝑔,𝐺,𝑢,𝑥   𝑓,𝐺,𝑦,𝑥   𝑥,𝑋,𝑓   𝜑,𝑥   𝑦,𝑔   𝜑,𝑓
Allowed substitution hints:   𝜑(𝑦,𝑢,𝑔)   𝐴(𝑦,𝑓,𝑔)   𝐶(𝑥,𝑦,𝑓)   𝐹(𝑥,𝑦,𝑢,𝑓,𝑔)   𝑋(𝑦,𝑢,𝑔)

Proof of Theorem tfr1onlemaccex
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑟 𝑠 𝑡 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfr1on.x . . 3 (𝜑 → Ord 𝑋)
2 ordelon 4173 . . 3 ((Ord 𝑋𝐶𝑋) → 𝐶 ∈ On)
31, 2sylan 277 . 2 ((𝜑𝐶𝑋) → 𝐶 ∈ On)
4 eleq1 2145 . . . . 5 (𝑧 = 𝑤 → (𝑧𝑋𝑤𝑋))
54anbi2d 452 . . . 4 (𝑧 = 𝑤 → ((𝜑𝑧𝑋) ↔ (𝜑𝑤𝑋)))
6 fneq2 5054 . . . . . 6 (𝑧 = 𝑤 → (𝑔 Fn 𝑧𝑔 Fn 𝑤))
7 raleq 2555 . . . . . 6 (𝑧 = 𝑤 → (∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
86, 7anbi12d 457 . . . . 5 (𝑧 = 𝑤 → ((𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
98exbidv 1748 . . . 4 (𝑧 = 𝑤 → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
105, 9imbi12d 232 . . 3 (𝑧 = 𝑤 → (((𝜑𝑧𝑋) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ ((𝜑𝑤𝑋) → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
11 eleq1 2145 . . . . 5 (𝑧 = 𝐶 → (𝑧𝑋𝐶𝑋))
1211anbi2d 452 . . . 4 (𝑧 = 𝐶 → ((𝜑𝑧𝑋) ↔ (𝜑𝐶𝑋)))
13 fneq2 5054 . . . . . 6 (𝑧 = 𝐶 → (𝑔 Fn 𝑧𝑔 Fn 𝐶))
14 raleq 2555 . . . . . 6 (𝑧 = 𝐶 → (∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
1513, 14anbi12d 457 . . . . 5 (𝑧 = 𝐶 → ((𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
1615exbidv 1748 . . . 4 (𝑧 = 𝐶 → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
1712, 16imbi12d 232 . . 3 (𝑧 = 𝐶 → (((𝜑𝑧𝑋) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ ((𝜑𝐶𝑋) → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
18 tfr1on.f . . . . . . . . 9 𝐹 = recs(𝐺)
19 tfr1on.g . . . . . . . . . 10 (𝜑 → Fun 𝐺)
2019ad3antrrr 476 . . . . . . . . 9 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → Fun 𝐺)
211ad3antrrr 476 . . . . . . . . 9 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → Ord 𝑋)
22 tfr1on.ex . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
23223expia 1141 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
2423alrimiv 1797 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
25 fneq1 5053 . . . . . . . . . . . . . . . . 17 (𝑓 = → (𝑓 Fn 𝑥 Fn 𝑥))
26 fveq2 5251 . . . . . . . . . . . . . . . . . 18 (𝑓 = → (𝐺𝑓) = (𝐺))
2726eleq1d 2151 . . . . . . . . . . . . . . . . 17 (𝑓 = → ((𝐺𝑓) ∈ V ↔ (𝐺) ∈ V))
2825, 27imbi12d 232 . . . . . . . . . . . . . . . 16 (𝑓 = → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ ( Fn 𝑥 → (𝐺) ∈ V)))
2928cbvalv 1837 . . . . . . . . . . . . . . 15 (∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ ∀( Fn 𝑥 → (𝐺) ∈ V))
3024, 29sylib 120 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → ∀( Fn 𝑥 → (𝐺) ∈ V))
313019.21bi 1491 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → ( Fn 𝑥 → (𝐺) ∈ V))
32313impia 1136 . . . . . . . . . . . 12 ((𝜑𝑥𝑋 Fn 𝑥) → (𝐺) ∈ V)
33323adant1r 1163 . . . . . . . . . . 11 (((𝜑𝑧 ∈ On) ∧ 𝑥𝑋 Fn 𝑥) → (𝐺) ∈ V)
34333adant1r 1163 . . . . . . . . . 10 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑥𝑋 Fn 𝑥) → (𝐺) ∈ V)
35343adant1r 1163 . . . . . . . . 9 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑥𝑋 Fn 𝑥) → (𝐺) ∈ V)
36 tfr1onlemsucfn.1 . . . . . . . . . 10 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
37 fveq1 5250 . . . . . . . . . . . . . . 15 (𝑓 = → (𝑓𝑦) = (𝑦))
38 reseq1 4663 . . . . . . . . . . . . . . . 16 (𝑓 = → (𝑓𝑦) = (𝑦))
3938fveq2d 5255 . . . . . . . . . . . . . . 15 (𝑓 = → (𝐺‘(𝑓𝑦)) = (𝐺‘(𝑦)))
4037, 39eqeq12d 2097 . . . . . . . . . . . . . 14 (𝑓 = → ((𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ (𝑦) = (𝐺‘(𝑦))))
4140ralbidv 2374 . . . . . . . . . . . . 13 (𝑓 = → (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)) ↔ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦))))
4225, 41anbi12d 457 . . . . . . . . . . . 12 (𝑓 = → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ ( Fn 𝑥 ∧ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦)))))
4342rexbidv 2375 . . . . . . . . . . 11 (𝑓 = → (∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦))) ↔ ∃𝑥𝑋 ( Fn 𝑥 ∧ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦)))))
4443cbvabv 2206 . . . . . . . . . 10 {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = { ∣ ∃𝑥𝑋 ( Fn 𝑥 ∧ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦)))}
4536, 44eqtri 2103 . . . . . . . . 9 𝐴 = { ∣ ∃𝑥𝑋 ( Fn 𝑥 ∧ ∀𝑦𝑥 (𝑦) = (𝐺‘(𝑦)))}
46 fneq1 5053 . . . . . . . . . . . . . . 15 (𝑟 = 𝑎 → (𝑟 Fn 𝑡𝑎 Fn 𝑡))
47 eleq1 2145 . . . . . . . . . . . . . . 15 (𝑟 = 𝑎 → (𝑟𝐴𝑎𝐴))
48 id 19 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑎𝑟 = 𝑎)
49 fveq2 5251 . . . . . . . . . . . . . . . . . . 19 (𝑟 = 𝑎 → (𝐺𝑟) = (𝐺𝑎))
5049opeq2d 3603 . . . . . . . . . . . . . . . . . 18 (𝑟 = 𝑎 → ⟨𝑡, (𝐺𝑟)⟩ = ⟨𝑡, (𝐺𝑎)⟩)
5150sneqd 3435 . . . . . . . . . . . . . . . . 17 (𝑟 = 𝑎 → {⟨𝑡, (𝐺𝑟)⟩} = {⟨𝑡, (𝐺𝑎)⟩})
5248, 51uneq12d 3139 . . . . . . . . . . . . . . . 16 (𝑟 = 𝑎 → (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩}) = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩}))
5352eqeq2d 2094 . . . . . . . . . . . . . . 15 (𝑟 = 𝑎 → (𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩}) ↔ 𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})))
5446, 47, 533anbi123d 1244 . . . . . . . . . . . . . 14 (𝑟 = 𝑎 → ((𝑟 Fn 𝑡𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩})) ↔ (𝑎 Fn 𝑡𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩}))))
5554cbvexv 1838 . . . . . . . . . . . . 13 (∃𝑟(𝑟 Fn 𝑡𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩})) ↔ ∃𝑎(𝑎 Fn 𝑡𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})))
5655rexbii 2379 . . . . . . . . . . . 12 (∃𝑡𝑧𝑟(𝑟 Fn 𝑡𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩})) ↔ ∃𝑡𝑧𝑎(𝑎 Fn 𝑡𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})))
57 fneq2 5054 . . . . . . . . . . . . . . 15 (𝑡 = 𝑏 → (𝑎 Fn 𝑡𝑎 Fn 𝑏))
58 opeq1 3596 . . . . . . . . . . . . . . . . . 18 (𝑡 = 𝑏 → ⟨𝑡, (𝐺𝑎)⟩ = ⟨𝑏, (𝐺𝑎)⟩)
5958sneqd 3435 . . . . . . . . . . . . . . . . 17 (𝑡 = 𝑏 → {⟨𝑡, (𝐺𝑎)⟩} = {⟨𝑏, (𝐺𝑎)⟩})
6059uneq2d 3138 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑏 → (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩}) = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))
6160eqeq2d 2094 . . . . . . . . . . . . . . 15 (𝑡 = 𝑏 → (𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩}) ↔ 𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})))
6257, 613anbi13d 1246 . . . . . . . . . . . . . 14 (𝑡 = 𝑏 → ((𝑎 Fn 𝑡𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})) ↔ (𝑎 Fn 𝑏𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
6362exbidv 1748 . . . . . . . . . . . . 13 (𝑡 = 𝑏 → (∃𝑎(𝑎 Fn 𝑡𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})) ↔ ∃𝑎(𝑎 Fn 𝑏𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
6463cbvrexv 2584 . . . . . . . . . . . 12 (∃𝑡𝑧𝑎(𝑎 Fn 𝑡𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑡, (𝐺𝑎)⟩})) ↔ ∃𝑏𝑧𝑎(𝑎 Fn 𝑏𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})))
6556, 64bitri 182 . . . . . . . . . . 11 (∃𝑡𝑧𝑟(𝑟 Fn 𝑡𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩})) ↔ ∃𝑏𝑧𝑎(𝑎 Fn 𝑏𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})))
6665abbii 2198 . . . . . . . . . 10 {𝑠 ∣ ∃𝑡𝑧𝑟(𝑟 Fn 𝑡𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩}))} = {𝑠 ∣ ∃𝑏𝑧𝑎(𝑎 Fn 𝑏𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))}
67 eqeq1 2089 . . . . . . . . . . . . . 14 (𝑠 = 𝑑 → (𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}) ↔ 𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})))
68673anbi3d 1250 . . . . . . . . . . . . 13 (𝑠 = 𝑑 → ((𝑎 Fn 𝑏𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})) ↔ (𝑎 Fn 𝑏𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
6968exbidv 1748 . . . . . . . . . . . 12 (𝑠 = 𝑑 → (∃𝑎(𝑎 Fn 𝑏𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})) ↔ ∃𝑎(𝑎 Fn 𝑏𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
7069rexbidv 2375 . . . . . . . . . . 11 (𝑠 = 𝑑 → (∃𝑏𝑧𝑎(𝑎 Fn 𝑏𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩})) ↔ ∃𝑏𝑧𝑎(𝑎 Fn 𝑏𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))))
7170cbvabv 2206 . . . . . . . . . 10 {𝑠 ∣ ∃𝑏𝑧𝑎(𝑎 Fn 𝑏𝑎𝐴𝑠 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))} = {𝑑 ∣ ∃𝑏𝑧𝑎(𝑎 Fn 𝑏𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))}
7266, 71eqtri 2103 . . . . . . . . 9 {𝑠 ∣ ∃𝑡𝑧𝑟(𝑟 Fn 𝑡𝑟𝐴𝑠 = (𝑟 ∪ {⟨𝑡, (𝐺𝑟)⟩}))} = {𝑑 ∣ ∃𝑏𝑧𝑎(𝑎 Fn 𝑏𝑎𝐴𝑑 = (𝑎 ∪ {⟨𝑏, (𝐺𝑎)⟩}))}
73 tfr1onlemaccex.u . . . . . . . . . . . 12 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
7473adantlr 461 . . . . . . . . . . 11 (((𝜑𝑧 ∈ On) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
7574adantlr 461 . . . . . . . . . 10 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
7675adantlr 461 . . . . . . . . 9 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑥 𝑋) → suc 𝑥𝑋)
77 simpr 108 . . . . . . . . 9 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → 𝑧𝑋)
78 simpr 108 . . . . . . . . . . . 12 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → 𝑏𝑧)
79 simplr 497 . . . . . . . . . . . 12 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → 𝑧𝑋)
80 ordtr1 4178 . . . . . . . . . . . . . 14 (Ord 𝑋 → ((𝑏𝑧𝑧𝑋) → 𝑏𝑋))
811, 80syl 14 . . . . . . . . . . . . 13 (𝜑 → ((𝑏𝑧𝑧𝑋) → 𝑏𝑋))
8281ad4antr 478 . . . . . . . . . . . 12 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → ((𝑏𝑧𝑧𝑋) → 𝑏𝑋))
8378, 79, 82mp2and 424 . . . . . . . . . . 11 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → 𝑏𝑋)
84 eleq1 2145 . . . . . . . . . . . . . 14 (𝑤 = 𝑏 → (𝑤𝑋𝑏𝑋))
85 fneq2 5054 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑏 → (𝑔 Fn 𝑤𝑔 Fn 𝑏))
86 raleq 2555 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑏 → (∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
8785, 86anbi12d 457 . . . . . . . . . . . . . . 15 (𝑤 = 𝑏 → ((𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑔 Fn 𝑏 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
8887exbidv 1748 . . . . . . . . . . . . . 14 (𝑤 = 𝑏 → (∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑔(𝑔 Fn 𝑏 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
8984, 88imbi12d 232 . . . . . . . . . . . . 13 (𝑤 = 𝑏 → ((𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ (𝑏𝑋 → ∃𝑔(𝑔 Fn 𝑏 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
90 simpllr 501 . . . . . . . . . . . . 13 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
9189, 90, 78rspcdva 2717 . . . . . . . . . . . 12 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → (𝑏𝑋 → ∃𝑔(𝑔 Fn 𝑏 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
92 fneq1 5053 . . . . . . . . . . . . . . 15 (𝑔 = 𝑎 → (𝑔 Fn 𝑏𝑎 Fn 𝑏))
93 fveq1 5250 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑎 → (𝑔𝑢) = (𝑎𝑢))
94 reseq1 4663 . . . . . . . . . . . . . . . . . 18 (𝑔 = 𝑎 → (𝑔𝑢) = (𝑎𝑢))
9594fveq2d 5255 . . . . . . . . . . . . . . . . 17 (𝑔 = 𝑎 → (𝐺‘(𝑔𝑢)) = (𝐺‘(𝑎𝑢)))
9693, 95eqeq12d 2097 . . . . . . . . . . . . . . . 16 (𝑔 = 𝑎 → ((𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ (𝑎𝑢) = (𝐺‘(𝑎𝑢))))
9796ralbidv 2374 . . . . . . . . . . . . . . 15 (𝑔 = 𝑎 → (∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢))))
9892, 97anbi12d 457 . . . . . . . . . . . . . 14 (𝑔 = 𝑎 → ((𝑔 Fn 𝑏 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ (𝑎 Fn 𝑏 ∧ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢)))))
9998cbvexv 1838 . . . . . . . . . . . . 13 (∃𝑔(𝑔 Fn 𝑏 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑎(𝑎 Fn 𝑏 ∧ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢))))
100 fveq2 5251 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑐 → (𝑎𝑢) = (𝑎𝑐))
101 reseq2 4664 . . . . . . . . . . . . . . . . . 18 (𝑢 = 𝑐 → (𝑎𝑢) = (𝑎𝑐))
102101fveq2d 5255 . . . . . . . . . . . . . . . . 17 (𝑢 = 𝑐 → (𝐺‘(𝑎𝑢)) = (𝐺‘(𝑎𝑐)))
103100, 102eqeq12d 2097 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑐 → ((𝑎𝑢) = (𝐺‘(𝑎𝑢)) ↔ (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
104103cbvralv 2583 . . . . . . . . . . . . . . 15 (∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢)) ↔ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))
105104anbi2i 445 . . . . . . . . . . . . . 14 ((𝑎 Fn 𝑏 ∧ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢))) ↔ (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
106105exbii 1537 . . . . . . . . . . . . 13 (∃𝑎(𝑎 Fn 𝑏 ∧ ∀𝑢𝑏 (𝑎𝑢) = (𝐺‘(𝑎𝑢))) ↔ ∃𝑎(𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
10799, 106bitri 182 . . . . . . . . . . . 12 (∃𝑔(𝑔 Fn 𝑏 ∧ ∀𝑢𝑏 (𝑔𝑢) = (𝐺‘(𝑔𝑢))) ↔ ∃𝑎(𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
10891, 107syl6ib 159 . . . . . . . . . . 11 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → (𝑏𝑋 → ∃𝑎(𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))))
10983, 108mpd 13 . . . . . . . . . 10 (((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) ∧ 𝑏𝑧) → ∃𝑎(𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
110109ralrimiva 2440 . . . . . . . . 9 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → ∀𝑏𝑧𝑎(𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐))))
11118, 20, 21, 35, 45, 72, 76, 77, 110tfr1onlemex 6042 . . . . . . . 8 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → ∃( Fn 𝑧 ∧ ∀𝑢𝑧 (𝑢) = (𝐺‘(𝑢))))
112 fneq1 5053 . . . . . . . . . 10 ( = 𝑔 → ( Fn 𝑧𝑔 Fn 𝑧))
113 fveq1 5250 . . . . . . . . . . . 12 ( = 𝑔 → (𝑢) = (𝑔𝑢))
114 reseq1 4663 . . . . . . . . . . . . 13 ( = 𝑔 → (𝑢) = (𝑔𝑢))
115114fveq2d 5255 . . . . . . . . . . . 12 ( = 𝑔 → (𝐺‘(𝑢)) = (𝐺‘(𝑔𝑢)))
116113, 115eqeq12d 2097 . . . . . . . . . . 11 ( = 𝑔 → ((𝑢) = (𝐺‘(𝑢)) ↔ (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
117116ralbidv 2374 . . . . . . . . . 10 ( = 𝑔 → (∀𝑢𝑧 (𝑢) = (𝐺‘(𝑢)) ↔ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
118112, 117anbi12d 457 . . . . . . . . 9 ( = 𝑔 → (( Fn 𝑧 ∧ ∀𝑢𝑧 (𝑢) = (𝐺‘(𝑢))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
119118cbvexv 1838 . . . . . . . 8 (∃( Fn 𝑧 ∧ ∀𝑢𝑧 (𝑢) = (𝐺‘(𝑢))) ↔ ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
120111, 119sylib 120 . . . . . . 7 ((((𝜑𝑧 ∈ On) ∧ ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ∧ 𝑧𝑋) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
121120exp31 356 . . . . . 6 ((𝜑𝑧 ∈ On) → (∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (𝑧𝑋 → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
122121expcom 114 . . . . 5 (𝑧 ∈ On → (𝜑 → (∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → (𝑧𝑋 → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))))
123122a2d 26 . . . 4 (𝑧 ∈ On → ((𝜑 → ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) → (𝜑 → (𝑧𝑋 → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))))
124 impexp 259 . . . . . 6 (((𝜑𝑤𝑋) → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ (𝜑 → (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
125124ralbii 2378 . . . . 5 (∀𝑤𝑧 ((𝜑𝑤𝑋) → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ ∀𝑤𝑧 (𝜑 → (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
126 r19.21v 2444 . . . . 5 (∀𝑤𝑧 (𝜑 → (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) ↔ (𝜑 → ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
127125, 126bitri 182 . . . 4 (∀𝑤𝑧 ((𝜑𝑤𝑋) → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ (𝜑 → ∀𝑤𝑧 (𝑤𝑋 → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
128 impexp 259 . . . 4 (((𝜑𝑧𝑋) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) ↔ (𝜑 → (𝑧𝑋 → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
129123, 127, 1283imtr4g 203 . . 3 (𝑧 ∈ On → (∀𝑤𝑧 ((𝜑𝑤𝑋) → ∃𝑔(𝑔 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))) → ((𝜑𝑧𝑋) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))))
13010, 17, 129tfis3 4363 . 2 (𝐶 ∈ On → ((𝜑𝐶𝑋) → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
1313, 130mpcom 36 1 ((𝜑𝐶𝑋) → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920  wal 1283   = wceq 1285  wex 1422  wcel 1434  {cab 2069  wral 2353  wrex 2354  Vcvv 2612  cun 2982  {csn 3422  cop 3425   cuni 3627  Ord word 4152  Oncon0 4153  suc csuc 4155  cres 4401  Fun wfun 4961   Fn wfn 4962  cfv 4967  recscrecs 5999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-suc 4161  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-recs 6000
This theorem is referenced by:  tfr1onlemres  6044
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