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Theorem tfrlemi1 5977
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 107 . . . . . . 7 ((𝑧 = 𝑤𝑔 = 𝑘) → 𝑔 = 𝑘)
2 simpl 106 . . . . . . 7 ((𝑧 = 𝑤𝑔 = 𝑘) → 𝑧 = 𝑤)
31, 2fneq12d 5019 . . . . . 6 ((𝑧 = 𝑤𝑔 = 𝑘) → (𝑔 Fn 𝑧𝑘 Fn 𝑤))
41fveq1d 5208 . . . . . . . 8 ((𝑧 = 𝑤𝑔 = 𝑘) → (𝑔𝑢) = (𝑘𝑢))
51reseq1d 4639 . . . . . . . . 9 ((𝑧 = 𝑤𝑔 = 𝑘) → (𝑔𝑢) = (𝑘𝑢))
65fveq2d 5210 . . . . . . . 8 ((𝑧 = 𝑤𝑔 = 𝑘) → (𝐹‘(𝑔𝑢)) = (𝐹‘(𝑘𝑢)))
74, 6eqeq12d 2070 . . . . . . 7 ((𝑧 = 𝑤𝑔 = 𝑘) → ((𝑔𝑢) = (𝐹‘(𝑔𝑢)) ↔ (𝑘𝑢) = (𝐹‘(𝑘𝑢))))
82, 7raleqbidv 2534 . . . . . 6 ((𝑧 = 𝑤𝑔 = 𝑘) → (∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)) ↔ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))
93, 8anbi12d 450 . . . . 5 ((𝑧 = 𝑤𝑔 = 𝑘) → ((𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ (𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))))
109cbvexdva 1820 . . . 4 (𝑧 = 𝑤 → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ ∃𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))))
1110imbi2d 223 . . 3 (𝑧 = 𝑤 → ((𝜑 → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) ↔ (𝜑 → ∃𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))))
12 fneq2 5016 . . . . . 6 (𝑧 = 𝐶 → (𝑔 Fn 𝑧𝑔 Fn 𝐶))
13 raleq 2522 . . . . . 6 (𝑧 = 𝐶 → (∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)) ↔ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
1412, 13anbi12d 450 . . . . 5 (𝑧 = 𝐶 → ((𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ (𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
1514exbidv 1722 . . . 4 (𝑧 = 𝐶 → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
1615imbi2d 223 . . 3 (𝑧 = 𝐶 → ((𝜑 → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) ↔ (𝜑 → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))))
17 r19.21v 2413 . . . 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 5957 . . . . . . . 8 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑒𝑧 (𝑔𝑒) = (𝐹‘(𝑔𝑒)))}
20 tfrlemisucfn.2 . . . . . . . . . 10 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
21 fveq2 5206 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
2221eleq1d 2122 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝐹𝑥) ∈ V ↔ (𝐹𝑧) ∈ V))
2322anbi2d 445 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹𝑧) ∈ V)))
2423cbvalv 1810 . . . . . . . . . 10 (∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
2520, 24sylib 131 . . . . . . . . 9 (𝜑 → ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
2625adantr 265 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))) → ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
27 simpr 107 . . . . . . . . . . . . 13 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → 𝑘 = 𝑓)
28 simplr 490 . . . . . . . . . . . . 13 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → 𝑤 = 𝑣)
2927, 28fneq12d 5019 . . . . . . . . . . . 12 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝑘 Fn 𝑤𝑓 Fn 𝑣))
3027eleq1d 2122 . . . . . . . . . . . 12 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝑘𝐴𝑓𝐴))
31 simpll 489 . . . . . . . . . . . . 13 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → 𝑡 = )
3227fveq2d 5210 . . . . . . . . . . . . . . . 16 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝐹𝑘) = (𝐹𝑓))
3328, 32opeq12d 3585 . . . . . . . . . . . . . . 15 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → ⟨𝑤, (𝐹𝑘)⟩ = ⟨𝑣, (𝐹𝑓)⟩)
3433sneqd 3416 . . . . . . . . . . . . . 14 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → {⟨𝑤, (𝐹𝑘)⟩} = {⟨𝑣, (𝐹𝑓)⟩})
3527, 34uneq12d 3126 . . . . . . . . . . . . 13 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩}) = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))
3631, 35eqeq12d 2070 . . . . . . . . . . . 12 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩}) ↔ = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩})))
3729, 30, 363anbi123d 1218 . . . . . . . . . . 11 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → ((𝑘 Fn 𝑤𝑘𝐴𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩})) ↔ (𝑓 Fn 𝑣𝑓𝐴 = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))))
3837cbvexdva 1820 . . . . . . . . . 10 ((𝑡 = 𝑤 = 𝑣) → (∃𝑘(𝑘 Fn 𝑤𝑘𝐴𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩})) ↔ ∃𝑓(𝑓 Fn 𝑣𝑓𝐴 = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))))
3938cbvrexdva 2557 . . . . . . . . 9 (𝑡 = → (∃𝑤𝑧𝑘(𝑘 Fn 𝑤𝑘𝐴𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩})) ↔ ∃𝑣𝑧𝑓(𝑓 Fn 𝑣𝑓𝐴 = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))))
4039cbvabv 2177 . . . . . . . 8 {𝑡 ∣ ∃𝑤𝑧𝑘(𝑘 Fn 𝑤𝑘𝐴𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩}))} = { ∣ ∃𝑣𝑧𝑓(𝑓 Fn 𝑣𝑓𝐴 = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))}
41 simpl 106 . . . . . . . . 9 ((𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))) → 𝑧 ∈ On)
4241adantl 266 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))) → 𝑧 ∈ On)
43 simpr 107 . . . . . . . . . 10 ((𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))) → ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))
44 simpr 107 . . . . . . . . . . . . . 14 ((𝑤 = 𝑣𝑘 = 𝑓) → 𝑘 = 𝑓)
45 simpl 106 . . . . . . . . . . . . . 14 ((𝑤 = 𝑣𝑘 = 𝑓) → 𝑤 = 𝑣)
4644, 45fneq12d 5019 . . . . . . . . . . . . 13 ((𝑤 = 𝑣𝑘 = 𝑓) → (𝑘 Fn 𝑤𝑓 Fn 𝑣))
47 simplr 490 . . . . . . . . . . . . . . . 16 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → 𝑘 = 𝑓)
48 simpr 107 . . . . . . . . . . . . . . . 16 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → 𝑢 = 𝑦)
4947, 48fveq12d 5212 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → (𝑘𝑢) = (𝑓𝑦))
5047, 48reseq12d 4641 . . . . . . . . . . . . . . . 16 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → (𝑘𝑢) = (𝑓𝑦))
5150fveq2d 5210 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → (𝐹‘(𝑘𝑢)) = (𝐹‘(𝑓𝑦)))
5249, 51eqeq12d 2070 . . . . . . . . . . . . . 14 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → ((𝑘𝑢) = (𝐹‘(𝑘𝑢)) ↔ (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
53 simpll 489 . . . . . . . . . . . . . 14 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → 𝑤 = 𝑣)
5452, 53cbvraldva2 2554 . . . . . . . . . . . . 13 ((𝑤 = 𝑣𝑘 = 𝑓) → (∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)) ↔ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
5546, 54anbi12d 450 . . . . . . . . . . . 12 ((𝑤 = 𝑣𝑘 = 𝑓) → ((𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))) ↔ (𝑓 Fn 𝑣 ∧ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
5655cbvexdva 1820 . . . . . . . . . . 11 (𝑤 = 𝑣 → (∃𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))) ↔ ∃𝑓(𝑓 Fn 𝑣 ∧ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
5756cbvralv 2550 . . . . . . . . . 10 (∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))) ↔ ∀𝑣𝑧𝑓(𝑓 Fn 𝑣 ∧ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
5843, 57sylib 131 . . . . . . . . 9 ((𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))) → ∀𝑣𝑧𝑓(𝑓 Fn 𝑣 ∧ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
5958adantl 266 . . . . . . . 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 5976 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
6160expr 361 . . . . . 6 ((𝜑𝑧 ∈ On) → (∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
6261expcom 113 . . . . 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 145 . . 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 4337 . 2 (𝐶 ∈ On → (𝜑 → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
6665impcom 120 1 ((𝜑𝐶 ∈ On) → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896  wal 1257   = wceq 1259  wex 1397  wcel 1409  {cab 2042  wral 2323  wrex 2324  Vcvv 2574  cun 2943  {csn 3403  cop 3406  Oncon0 4128  cres 4375  Fun wfun 4924   Fn wfn 4925  cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-recs 5951
This theorem is referenced by:  tfrlemi14d  5978  tfrexlem  5979
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