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Theorem tfrlemi1 5946
 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 103 . . . . . . 7 ((𝑧 = 𝑤𝑔 = 𝑘) → 𝑔 = 𝑘)
2 simpl 102 . . . . . . 7 ((𝑧 = 𝑤𝑔 = 𝑘) → 𝑧 = 𝑤)
31, 2fneq12d 4991 . . . . . 6 ((𝑧 = 𝑤𝑔 = 𝑘) → (𝑔 Fn 𝑧𝑘 Fn 𝑤))
41fveq1d 5180 . . . . . . . 8 ((𝑧 = 𝑤𝑔 = 𝑘) → (𝑔𝑢) = (𝑘𝑢))
51reseq1d 4611 . . . . . . . . 9 ((𝑧 = 𝑤𝑔 = 𝑘) → (𝑔𝑢) = (𝑘𝑢))
65fveq2d 5182 . . . . . . . 8 ((𝑧 = 𝑤𝑔 = 𝑘) → (𝐹‘(𝑔𝑢)) = (𝐹‘(𝑘𝑢)))
74, 6eqeq12d 2054 . . . . . . 7 ((𝑧 = 𝑤𝑔 = 𝑘) → ((𝑔𝑢) = (𝐹‘(𝑔𝑢)) ↔ (𝑘𝑢) = (𝐹‘(𝑘𝑢))))
82, 7raleqbidv 2517 . . . . . 6 ((𝑧 = 𝑤𝑔 = 𝑘) → (∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)) ↔ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))
93, 8anbi12d 442 . . . . 5 ((𝑧 = 𝑤𝑔 = 𝑘) → ((𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ (𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))))
109cbvexdva 1804 . . . 4 (𝑧 = 𝑤 → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ ∃𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))))
1110imbi2d 219 . . 3 (𝑧 = 𝑤 → ((𝜑 → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) ↔ (𝜑 → ∃𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))))
12 fneq2 4988 . . . . . 6 (𝑧 = 𝐶 → (𝑔 Fn 𝑧𝑔 Fn 𝐶))
13 raleq 2505 . . . . . 6 (𝑧 = 𝐶 → (∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)) ↔ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
1412, 13anbi12d 442 . . . . 5 (𝑧 = 𝐶 → ((𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ (𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
1514exbidv 1706 . . . 4 (𝑧 = 𝐶 → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))) ↔ ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
1615imbi2d 219 . . 3 (𝑧 = 𝐶 → ((𝜑 → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))) ↔ (𝜑 → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))))
17 r19.21v 2396 . . . 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 5926 . . . . . . . 8 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑒𝑧 (𝑔𝑒) = (𝐹‘(𝑔𝑒)))}
20 tfrlemisucfn.2 . . . . . . . . . 10 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
21 fveq2 5178 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
2221eleq1d 2106 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝐹𝑥) ∈ V ↔ (𝐹𝑧) ∈ V))
2322anbi2d 437 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ (Fun 𝐹 ∧ (𝐹𝑧) ∈ V)))
2423cbvalv 1794 . . . . . . . . . 10 (∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V) ↔ ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
2520, 24sylib 127 . . . . . . . . 9 (𝜑 → ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
2625adantr 261 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))) → ∀𝑧(Fun 𝐹 ∧ (𝐹𝑧) ∈ V))
27 simpr 103 . . . . . . . . . . . . 13 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → 𝑘 = 𝑓)
28 simplr 482 . . . . . . . . . . . . 13 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → 𝑤 = 𝑣)
2927, 28fneq12d 4991 . . . . . . . . . . . 12 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝑘 Fn 𝑤𝑓 Fn 𝑣))
3027eleq1d 2106 . . . . . . . . . . . 12 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝑘𝐴𝑓𝐴))
31 simpll 481 . . . . . . . . . . . . 13 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → 𝑡 = )
3227fveq2d 5182 . . . . . . . . . . . . . . . 16 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝐹𝑘) = (𝐹𝑓))
3328, 32opeq12d 3557 . . . . . . . . . . . . . . 15 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → ⟨𝑤, (𝐹𝑘)⟩ = ⟨𝑣, (𝐹𝑓)⟩)
3433sneqd 3388 . . . . . . . . . . . . . 14 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → {⟨𝑤, (𝐹𝑘)⟩} = {⟨𝑣, (𝐹𝑓)⟩})
3527, 34uneq12d 3098 . . . . . . . . . . . . 13 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩}) = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))
3631, 35eqeq12d 2054 . . . . . . . . . . . 12 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → (𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩}) ↔ = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩})))
3729, 30, 363anbi123d 1207 . . . . . . . . . . 11 (((𝑡 = 𝑤 = 𝑣) ∧ 𝑘 = 𝑓) → ((𝑘 Fn 𝑤𝑘𝐴𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩})) ↔ (𝑓 Fn 𝑣𝑓𝐴 = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))))
3837cbvexdva 1804 . . . . . . . . . 10 ((𝑡 = 𝑤 = 𝑣) → (∃𝑘(𝑘 Fn 𝑤𝑘𝐴𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩})) ↔ ∃𝑓(𝑓 Fn 𝑣𝑓𝐴 = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))))
3938cbvrexdva 2540 . . . . . . . . 9 (𝑡 = → (∃𝑤𝑧𝑘(𝑘 Fn 𝑤𝑘𝐴𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩})) ↔ ∃𝑣𝑧𝑓(𝑓 Fn 𝑣𝑓𝐴 = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))))
4039cbvabv 2161 . . . . . . . 8 {𝑡 ∣ ∃𝑤𝑧𝑘(𝑘 Fn 𝑤𝑘𝐴𝑡 = (𝑘 ∪ {⟨𝑤, (𝐹𝑘)⟩}))} = { ∣ ∃𝑣𝑧𝑓(𝑓 Fn 𝑣𝑓𝐴 = (𝑓 ∪ {⟨𝑣, (𝐹𝑓)⟩}))}
41 simpl 102 . . . . . . . . 9 ((𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))) → 𝑧 ∈ On)
4241adantl 262 . . . . . . . 8 ((𝜑 ∧ (𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))) → 𝑧 ∈ On)
43 simpr 103 . . . . . . . . . 10 ((𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))) → ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))
44 simpr 103 . . . . . . . . . . . . . 14 ((𝑤 = 𝑣𝑘 = 𝑓) → 𝑘 = 𝑓)
45 simpl 102 . . . . . . . . . . . . . 14 ((𝑤 = 𝑣𝑘 = 𝑓) → 𝑤 = 𝑣)
4644, 45fneq12d 4991 . . . . . . . . . . . . 13 ((𝑤 = 𝑣𝑘 = 𝑓) → (𝑘 Fn 𝑤𝑓 Fn 𝑣))
47 simplr 482 . . . . . . . . . . . . . . . 16 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → 𝑘 = 𝑓)
48 simpr 103 . . . . . . . . . . . . . . . 16 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → 𝑢 = 𝑦)
4947, 48fveq12d 5184 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → (𝑘𝑢) = (𝑓𝑦))
5047, 48reseq12d 4613 . . . . . . . . . . . . . . . 16 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → (𝑘𝑢) = (𝑓𝑦))
5150fveq2d 5182 . . . . . . . . . . . . . . 15 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → (𝐹‘(𝑘𝑢)) = (𝐹‘(𝑓𝑦)))
5249, 51eqeq12d 2054 . . . . . . . . . . . . . 14 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → ((𝑘𝑢) = (𝐹‘(𝑘𝑢)) ↔ (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
53 simpll 481 . . . . . . . . . . . . . 14 (((𝑤 = 𝑣𝑘 = 𝑓) ∧ 𝑢 = 𝑦) → 𝑤 = 𝑣)
5452, 53cbvraldva2 2537 . . . . . . . . . . . . 13 ((𝑤 = 𝑣𝑘 = 𝑓) → (∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)) ↔ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
5546, 54anbi12d 442 . . . . . . . . . . . 12 ((𝑤 = 𝑣𝑘 = 𝑓) → ((𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))) ↔ (𝑓 Fn 𝑣 ∧ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
5655cbvexdva 1804 . . . . . . . . . . 11 (𝑤 = 𝑣 → (∃𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))) ↔ ∃𝑓(𝑓 Fn 𝑣 ∧ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))))
5756cbvralv 2533 . . . . . . . . . 10 (∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))) ↔ ∀𝑣𝑧𝑓(𝑓 Fn 𝑣 ∧ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
5843, 57sylib 127 . . . . . . . . 9 ((𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))) → ∀𝑣𝑧𝑓(𝑓 Fn 𝑣 ∧ ∀𝑦𝑣 (𝑓𝑦) = (𝐹‘(𝑓𝑦))))
5958adantl 262 . . . . . . . 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 5945 . . . . . . 7 ((𝜑 ∧ (𝑧 ∈ On ∧ ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))))) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
6160expr 357 . . . . . 6 ((𝜑𝑧 ∈ On) → (∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
6261expcom 109 . . . . 5 (𝑧 ∈ On → (𝜑 → (∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢))) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))))
6362a2d 23 . . . 4 (𝑧 ∈ On → ((𝜑 → ∀𝑤𝑧𝑘(𝑘 Fn 𝑤 ∧ ∀𝑢𝑤 (𝑘𝑢) = (𝐹‘(𝑘𝑢)))) → (𝜑 → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))))
6417, 63syl5bi 141 . . 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 4309 . 2 (𝐶 ∈ On → (𝜑 → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))))
6665impcom 116 1 ((𝜑𝐶 ∈ On) → ∃𝑔(𝑔 Fn 𝐶 ∧ ∀𝑢𝐶 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  Vcvv 2557   ∪ cun 2915  {csn 3375  ⟨cop 3378  Oncon0 4100   ↾ cres 4347  Fun wfun 4896   Fn wfn 4897  ‘cfv 4902 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920 This theorem is referenced by:  tfrlemi14d  5947  tfrexlem  5948
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