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Theorem frrlem11 31520
Description: Lemma for founded recursion. Here, we calculate the value of 𝐹 (the union of all acceptable functions). (Contributed by Paul Chapman, 21-Apr-2012.)
Hypotheses
Ref Expression
frrlem10.1 𝑅 Fr 𝐴
frrlem10.2 𝑅 Se 𝐴
frrlem10.3 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
frrlem10.4 𝐹 = 𝐵
Assertion
Ref Expression
frrlem11 (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝑥,𝐹   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦   𝑥,𝐵   𝑓,𝐹
Allowed substitution hints:   𝐵(𝑦,𝑓)   𝐹(𝑦)

Proof of Theorem frrlem11
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 vex 3192 . . 3 𝑦 ∈ V
21eldm2 5287 . 2 (𝑦 ∈ dom 𝐹 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐹)
3 frrlem10.4 . . . . . . 7 𝐹 = 𝐵
4 frrlem10.3 . . . . . . . 8 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
54unieqi 4416 . . . . . . 7 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
63, 5eqtri 2643 . . . . . 6 𝐹 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
76eleq2i 2690 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑧⟩ ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))})
8 eluniab 4418 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))} ↔ ∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))))
97, 8bitri 264 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝐹 ↔ ∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))))
104abeq2i 2732 . . . . . . . 8 (𝑓𝐵 ↔ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))))
11 elssuni 4438 . . . . . . . . 9 (𝑓𝐵𝑓 𝐵)
1211, 3syl6sseqr 3636 . . . . . . . 8 (𝑓𝐵𝑓𝐹)
1310, 12sylbir 225 . . . . . . 7 (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝑓𝐹)
14 fnop 5957 . . . . . . . . . . 11 ((𝑓 Fn 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) → 𝑦𝑥)
1514ex 450 . . . . . . . . . 10 (𝑓 Fn 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝑓𝑦𝑥))
16 rsp 2924 . . . . . . . . . . . . . . . . . . 19 (∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑦𝑥 → (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
1716impcom 446 . . . . . . . . . . . . . . . . . 18 ((𝑦𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
18 rsp 2924 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦𝑥 → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
19 fndm 5953 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
20 sseq2 3611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (dom 𝑓 = 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
21 eleq2 2687 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (dom 𝑓 = 𝑥 → (𝑦 ∈ dom 𝑓𝑦𝑥))
2220, 21anbi12d 746 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (dom 𝑓 = 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥𝑦𝑥)))
2319, 22syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓 Fn 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥𝑦𝑥)))
2423biimprd 238 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 Fn 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥𝑦𝑥) → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)))
2524expd 452 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 Fn 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓))))
2625impcom 446 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥𝑓 Fn 𝑥) → (𝑦𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)))
27 frrlem10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑅 Fr 𝐴
28 frrlem10.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑅 Se 𝐴
2927, 28, 4, 3frrlem10 31519 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Fun 𝐹
30 funssfv 6171 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((Fun 𝐹𝑓𝐹𝑦 ∈ dom 𝑓) → (𝐹𝑦) = (𝑓𝑦))
31303adant3l 1319 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((Fun 𝐹𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → (𝐹𝑦) = (𝑓𝑦))
32 fun2ssres 5894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((Fun 𝐹𝑓𝐹 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
33323adant3r 1320 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((Fun 𝐹𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
3433oveq2d 6626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((Fun 𝐹𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
3531, 34eqeq12d 2636 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((Fun 𝐹𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → ((𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
3635biimprd 238 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Fun 𝐹𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → ((𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
3729, 36mp3an1 1408 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → ((𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
3837expcom 451 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓) → (𝑓𝐹 → ((𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
3938com23 86 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓) → ((𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
4026, 39syl6com 37 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥𝑓 Fn 𝑥) → ((𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))))
4140expd 452 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑓 Fn 𝑥 → ((𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4241com34 91 . . . . . . . . . . . . . . . . . . . . 21 (𝑦𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → ((𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓 Fn 𝑥 → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4318, 42sylcom 30 . . . . . . . . . . . . . . . . . . . 20 (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦𝑥 → ((𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓 Fn 𝑥 → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4443adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑦𝑥 → ((𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓 Fn 𝑥 → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4544com14 96 . . . . . . . . . . . . . . . . . 18 (𝑓 Fn 𝑥 → (𝑦𝑥 → ((𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4617, 45syl7 74 . . . . . . . . . . . . . . . . 17 (𝑓 Fn 𝑥 → (𝑦𝑥 → ((𝑦𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4746exp4a 632 . . . . . . . . . . . . . . . 16 (𝑓 Fn 𝑥 → (𝑦𝑥 → (𝑦𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))))))
4847pm2.43d 53 . . . . . . . . . . . . . . 15 (𝑓 Fn 𝑥 → (𝑦𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4948com34 91 . . . . . . . . . . . . . 14 (𝑓 Fn 𝑥 → (𝑦𝑥 → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
5049imp 445 . . . . . . . . . . . . 13 ((𝑓 Fn 𝑥𝑦𝑥) → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))))
5150expd 452 . . . . . . . . . . . 12 ((𝑓 Fn 𝑥𝑦𝑥) → (𝑥𝐴 → (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
52513impd 1278 . . . . . . . . . . 11 ((𝑓 Fn 𝑥𝑦𝑥) → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
5352ex 450 . . . . . . . . . 10 (𝑓 Fn 𝑥 → (𝑦𝑥 → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))))
5415, 53syldc 48 . . . . . . . . 9 (⟨𝑦, 𝑧⟩ ∈ 𝑓 → (𝑓 Fn 𝑥 → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))))
5554impd 447 . . . . . . . 8 (⟨𝑦, 𝑧⟩ ∈ 𝑓 → ((𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
5655exlimdv 1858 . . . . . . 7 (⟨𝑦, 𝑧⟩ ∈ 𝑓 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝑓𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
5713, 56mpdi 45 . . . . . 6 (⟨𝑦, 𝑧⟩ ∈ 𝑓 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
5857imp 445 . . . . 5 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
5958exlimiv 1855 . . . 4 (∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))) → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
609, 59sylbi 207 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
6160exlimiv 1855 . 2 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
622, 61sylbi 207 1 (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝑦𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wss 3559  cop 4159   cuni 4407   Fr wfr 5035   Se wse 5036  dom cdm 5079  cres 5081  Predcpred 5643  Fun wfun 5846   Fn wfn 5847  cfv 5852  (class class class)co 6610
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489
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-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-trpred 31446
This theorem is referenced by: (None)
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