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Theorem tfrlemi1 6053
Description: We can define an acceptable function on any ordinal.

As with many of the transfinite recursion theorems, we have a hypothesis that states that 𝐹 is a function and that it is defined for all ordinals. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Hypotheses
Ref Expression
tfrlemisucfn.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrlemisucfn.2 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
Assertion
Ref Expression
tfrlemi1 ((𝜑𝐶 ∈ On) → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
Distinct variable groups:   𝑓,𝑔,𝑢,𝑥,𝑦,𝐴   𝑓,𝐹,𝑔,𝑢,𝑥,𝑦   𝜑,𝑦   𝐶,𝑔,𝑢   𝜑,𝑓
Allowed substitution hints:   𝜑(𝑥,𝑢,𝑔)   𝐶(𝑥,𝑦,𝑓)

Proof of Theorem tfrlemi1
Dummy variables 𝑒 𝑘 𝑡 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 108 . . . . . . 7 ((𝑧 = 𝑤𝑔 = 𝑘) → 𝑔 = 𝑘)
2 simpl 107 . . . . . . 7 ((𝑧 = 𝑤𝑔 = 𝑘) → 𝑧 = 𝑤)
31, 2fneq12d 5073 . . . . . 6 ((𝑧 = 𝑤𝑔 = 𝑘) → (𝑔 Fn 𝑧𝑘 Fn 𝑤))
41fveq1d 5272 . . . . . . . 8 ((𝑧 = 𝑤𝑔 = 𝑘) → (𝑔𝑢) = (𝑘𝑢))
51reseq1d 4682 . . . . . . . . 9 ((𝑧 = 𝑤𝑔 = 𝑘) → (𝑔𝑢) = (𝑘𝑢))
65fveq2d 5274 . . . . . . . 8 ((𝑧 = 𝑤𝑔 = 𝑘) → (𝐹‘(𝑔𝑢)) = (𝐹‘(𝑘𝑢)))
74, 6eqeq12d 2099 . . . . . . 7 ((𝑧 = 𝑤𝑔 = 𝑘) → ((𝑔𝑢) = (𝐹‘(𝑔𝑢)) ↔ (𝑘𝑢) = (𝐹‘(𝑘𝑢))))
82, 7raleqbidv 2570 . . . . . 6 ((𝑧 = 𝑤𝑔 = 𝑘) → (∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)) ↔ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))
93, 8anbi12d 457 . . . . 5 ((𝑧 = 𝑤𝑔 = 𝑘) → ((𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ (𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))))
109cbvexdva 1849 . . . 4 (𝑧 = 𝑤 → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ ∃𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))))
1110imbi2d 228 . . 3 (𝑧 = 𝑤 → ((𝜑 → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) ↔ (𝜑 → ∃𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))))
12 fneq2 5070 . . . . . 6 (𝑧 = 𝐶 → (𝑔 Fn 𝑧𝑔 Fn 𝐶))
13 raleq 2558 . . . . . 6 (𝑧 = 𝐶 → (∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)) ↔ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
1412, 13anbi12d 457 . . . . 5 (𝑧 = 𝐶 → ((𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ (𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
1514exbidv 1750 . . . 4 (𝑧 = 𝐶 → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
1615imbi2d 228 . . 3 (𝑧 = 𝐶 → ((𝜑 → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) ↔ (𝜑 → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))))
17 r19.21v 2446 . . . 4 (∀𝑤𝑧 (𝜑 → ∃𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))) ↔ (𝜑 → ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))))
18 tfrlemisucfn.1 . . . . . . . . 9 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
1918tfrlem3 6032 . . . . . . . 8 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑒𝑧 (𝑔𝑒) = (𝐹‘(𝑔𝑒)))}
20 tfrlemisucfn.2 . . . . . . . . . 10 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
21 fveq2 5270 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
2221eleq1d 2153 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝐹𝑥) ∈ V ↔ (𝐹𝑧) ∈ V))
2322anbi2d 452 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹𝑧) ∈ V)))
2423cbvalv 1839 . . . . . . . . . 10 (∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
2520, 24sylib 120 . . . . . . . . 9 (𝜑 → ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
2625adantr 270 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))) → ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
27 simpr 108 . . . . . . . . . . . . 13 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → 𝑘 = 𝑓)
28 simplr 497 . . . . . . . . . . . . 13 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → 𝑤 = 𝑣)
2927, 28fneq12d 5073 . . . . . . . . . . . 12 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝑘 Fn 𝑤𝑓 Fn 𝑣))
3027eleq1d 2153 . . . . . . . . . . . 12 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝑘𝐴𝑓𝐴))
31 simpll 496 . . . . . . . . . . . . 13 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → 𝑡 = )
3227fveq2d 5274 . . . . . . . . . . . . . . . 16 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝐹𝑘) = (𝐹𝑓))
3328, 32opeq12d 3615 . . . . . . . . . . . . . . 15 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → ⟨𝑤, (𝐹𝑘)⟩ = ⟨𝑣, (𝐹𝑓)⟩)
3433sneqd 3444 . . . . . . . . . . . . . 14 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → {⟨𝑤, (𝐹𝑘)⟩} = {⟨𝑣, (𝐹𝑓)⟩})
3527, 34uneq12d 3144 . . . . . . . . . . . . 13 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩}) = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))
3631, 35eqeq12d 2099 . . . . . . . . . . . 12 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩}) ↔ = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩})))
3729, 30, 363anbi123d 1246 . . . . . . . . . . 11 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → ((𝑘 Fn 𝑤𝑘𝐴𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩})) ↔ (𝑓 Fn 𝑣𝑓𝐴 = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))))
3837cbvexdva 1849 . . . . . . . . . 10 ((𝑡 = 𝑤 = 𝑣) → (∃𝑘(𝑘 Fn 𝑤𝑘𝐴𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩})) ↔ ∃𝑓(𝑓 Fn 𝑣𝑓𝐴 = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))))
3938cbvrexdva 2593 . . . . . . . . 9 (𝑡 = → (∃𝑤𝑧𝑘(𝑘 Fn 𝑤𝑘𝐴𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩})) ↔ ∃𝑣𝑧𝑓(𝑓 Fn 𝑣𝑓𝐴 = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))))
4039cbvabv 2208 . . . . . . . 8 {𝑡 ∣ ∃𝑤𝑧𝑘(𝑘 Fn 𝑤𝑘𝐴𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩}))} = { ∣ ∃𝑣𝑧𝑓(𝑓 Fn 𝑣𝑓𝐴 = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))}
41 simpl 107 . . . . . . . . 9 ((𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))) → 𝑧 ∈ On)
4241adantl 271 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))) → 𝑧 ∈ On)
43 simpr 108 . . . . . . . . . 10 ((𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))) → ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))
44 simpr 108 . . . . . . . . . . . . . 14 ((𝑤 = 𝑣𝑘 = 𝑓) → 𝑘 = 𝑓)
45 simpl 107 . . . . . . . . . . . . . 14 ((𝑤 = 𝑣𝑘 = 𝑓) → 𝑤 = 𝑣)
4644, 45fneq12d 5073 . . . . . . . . . . . . 13 ((𝑤 = 𝑣𝑘 = 𝑓) → (𝑘 Fn 𝑤𝑓 Fn 𝑣))
47 simplr 497 . . . . . . . . . . . . . . . 16 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → 𝑘 = 𝑓)
48 simpr 108 . . . . . . . . . . . . . . . 16 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → 𝑢 = 𝑦)
4947, 48fveq12d 5277 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → (𝑘𝑢) = (𝑓𝑦))
5047, 48reseq12d 4684 . . . . . . . . . . . . . . . 16 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → (𝑘𝑢) = (𝑓𝑦))
5150fveq2d 5274 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → (𝐹‘(𝑘𝑢)) = (𝐹‘(𝑓𝑦)))
5249, 51eqeq12d 2099 . . . . . . . . . . . . . 14 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → ((𝑘𝑢) = (𝐹‘(𝑘𝑢)) ↔ (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
53 simpll 496 . . . . . . . . . . . . . 14 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → 𝑤 = 𝑣)
5452, 53cbvraldva2 2590 . . . . . . . . . . . . 13 ((𝑤 = 𝑣𝑘 = 𝑓) → (∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)) ↔ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
5546, 54anbi12d 457 . . . . . . . . . . . 12 ((𝑤 = 𝑣𝑘 = 𝑓) → ((𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))) ↔ (𝑓 Fn 𝑣 ∧ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
5655cbvexdva 1849 . . . . . . . . . . 11 (𝑤 = 𝑣 → (∃𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))) ↔ ∃𝑓(𝑓 Fn 𝑣 ∧ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
5756cbvralv 2586 . . . . . . . . . 10 (∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))) ↔ ∀𝑣𝑧𝑓(𝑓 Fn 𝑣 ∧ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
5843, 57sylib 120 . . . . . . . . 9 ((𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))) → ∀𝑣𝑧𝑓(𝑓 Fn 𝑣 ∧ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
5958adantl 271 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))) → ∀𝑣𝑧𝑓(𝑓 Fn 𝑣 ∧ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
6019, 26, 40, 42, 59tfrlemiex 6052 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
6160expr 367 . . . . . 6 ((𝜑𝑧 ∈ On) → (∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
6261expcom 114 . . . . 5 (𝑧 ∈ On → (𝜑 → (∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))))
6362a2d 26 . . . 4 (𝑧 ∈ On → ((𝜑 → ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))) → (𝜑 → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))))
6417, 63syl5bi 150 . . 3 (𝑧 ∈ On → (∀𝑤𝑧 (𝜑 → ∃𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))) → (𝜑 → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))))
6511, 16, 64tfis3 4376 . 2 (𝐶 ∈ On → (𝜑 → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
6665impcom 123 1 ((𝜑𝐶 ∈ On) → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 922  wal 1285   = wceq 1287  wex 1424  wcel 1436  {cab 2071  wral 2355  wrex 2356  Vcvv 2615  cun 2986  {csn 3431  cop 3434  Oncon0 4166  cres 4415  Fun wfun 4977   Fn wfn 4978  cfv 4983
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-id 4096  df-iord 4169  df-on 4171  df-suc 4174  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-recs 6026
This theorem is referenced by:  tfrlemi14d  6054  tfrexlem  6055
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