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Theorem wfrlem12 7956
 Description: Lemma for well-founded recursion. Here, we compute the value of the recursive definition generator. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfrfun.1 𝑅 We 𝐴
wfrfun.2 𝑅 Se 𝐴
wfrfun.3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
wfrlem12 (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐺   𝑦,𝑅
Allowed substitution hint:   𝐹(𝑦)

Proof of Theorem wfrlem12
Dummy variables 𝑓 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3444 . . 3 𝑦 ∈ V
21eldm2 5735 . 2 (𝑦 ∈ dom 𝐹 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐹)
3 wfrfun.3 . . . . . . 7 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4 df-wrecs 7937 . . . . . . 7 wrecs(𝑅, 𝐴, 𝐺) = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
53, 4eqtri 2821 . . . . . 6 𝐹 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
65eleq2i 2881 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑧⟩ ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
7 eluniab 4816 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))))
86, 7bitri 278 . . . 4 (⟨𝑦, 𝑧⟩ ∈ 𝐹 ↔ ∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))))
9 abid 2780 . . . . . . . 8 (𝑓 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
10 elssuni 4831 . . . . . . . . 9 (𝑓 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} → 𝑓 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
1110, 5sseqtrrdi 3966 . . . . . . . 8 (𝑓 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} → 𝑓𝐹)
129, 11sylbir 238 . . . . . . 7 (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → 𝑓𝐹)
13 fnop 6434 . . . . . . . . . . 11 ((𝑓 Fn 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) → 𝑦𝑥)
1413ex 416 . . . . . . . . . 10 (𝑓 Fn 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝑓𝑦𝑥))
15 rsp 3170 . . . . . . . . . . . . . . 15 (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑦𝑥 → (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
1615impcom 411 . . . . . . . . . . . . . 14 ((𝑦𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
17 rsp 3170 . . . . . . . . . . . . . . . . 17 (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦𝑥 → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
18 fndm 6428 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
1918sseq2d 3947 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 Fn 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
2018eleq2d 2875 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 Fn 𝑥 → (𝑦 ∈ dom 𝑓𝑦𝑥))
2119, 20anbi12d 633 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 Fn 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥𝑦𝑥)))
2221biimprd 251 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 Fn 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥𝑦𝑥) → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)))
2322expd 419 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 Fn 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓))))
2423impcom 411 . . . . . . . . . . . . . . . . . . . 20 ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥𝑓 Fn 𝑥) → (𝑦𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)))
25 wfrfun.1 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑅 We 𝐴
26 wfrfun.2 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑅 Se 𝐴
2725, 26, 3wfrfun 7955 . . . . . . . . . . . . . . . . . . . . . . 23 Fun 𝐹
28 funssfv 6671 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Fun 𝐹𝑓𝐹𝑦 ∈ dom 𝑓) → (𝐹𝑦) = (𝑓𝑦))
29283adant3l 1177 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Fun 𝐹𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → (𝐹𝑦) = (𝑓𝑦))
30 fun2ssres 6372 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Fun 𝐹𝑓𝐹 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
31303adant3r 1178 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Fun 𝐹𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
3231fveq2d 6654 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Fun 𝐹𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
3329, 32eqeq12d 2814 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Fun 𝐹𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → ((𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
3433biimprd 251 . . . . . . . . . . . . . . . . . . . . . . 23 ((Fun 𝐹𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → ((𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
3527, 34mp3an1 1445 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → ((𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
3635expcom 417 . . . . . . . . . . . . . . . . . . . . 21 ((Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓) → (𝑓𝐹 → ((𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
3736com23 86 . . . . . . . . . . . . . . . . . . . 20 ((Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓) → ((𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
3824, 37syl6com 37 . . . . . . . . . . . . . . . . . . 19 (𝑦𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥𝑓 Fn 𝑥) → ((𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))))
3938expd 419 . . . . . . . . . . . . . . . . . 18 (𝑦𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑓 Fn 𝑥 → ((𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4039com34 91 . . . . . . . . . . . . . . . . 17 (𝑦𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → ((𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓 Fn 𝑥 → (𝑓𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4117, 40sylcom 30 . . . . . . . . . . . . . . . 16 (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦𝑥 → ((𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓 Fn 𝑥 → (𝑓𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4241adantl 485 . . . . . . . . . . . . . . 15 ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑦𝑥 → ((𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓 Fn 𝑥 → (𝑓𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4342com14 96 . . . . . . . . . . . . . 14 (𝑓 Fn 𝑥 → (𝑦𝑥 → ((𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑓𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4416, 43syl7 74 . . . . . . . . . . . . 13 (𝑓 Fn 𝑥 → (𝑦𝑥 → ((𝑦𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑓𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4544exp4a 435 . . . . . . . . . . . 12 (𝑓 Fn 𝑥 → (𝑦𝑥 → (𝑦𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑓𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))))))
4645pm2.43d 53 . . . . . . . . . . 11 (𝑓 Fn 𝑥 → (𝑦𝑥 → (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑓𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4746com34 91 . . . . . . . . . 10 (𝑓 Fn 𝑥 → (𝑦𝑥 → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
4814, 47syldc 48 . . . . . . . . 9 (⟨𝑦, 𝑧⟩ ∈ 𝑓 → (𝑓 Fn 𝑥 → ((𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
49483impd 1345 . . . . . . . 8 (⟨𝑦, 𝑧⟩ ∈ 𝑓 → ((𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝑓𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
5049exlimdv 1934 . . . . . . 7 (⟨𝑦, 𝑧⟩ ∈ 𝑓 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝑓𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
5112, 50mpdi 45 . . . . . 6 (⟨𝑦, 𝑧⟩ ∈ 𝑓 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
5251imp 410 . . . . 5 ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
5352exlimiv 1931 . . . 4 (∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
548, 53sylbi 220 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
5554exlimiv 1931 . 2 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
562, 55sylbi 220 1 (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∀wral 3106   ⊆ wss 3881  ⟨cop 4531  ∪ cuni 4801   Se wse 5477   We wwe 5478  dom cdm 5520   ↾ cres 5522  Predcpred 6118  Fun wfun 6321   Fn wfn 6322  ‘cfv 6327  wrecscwrecs 7936 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-po 5439  df-so 5440  df-fr 5479  df-se 5480  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6119  df-iota 6286  df-fun 6329  df-fn 6330  df-fv 6335  df-wrecs 7937 This theorem is referenced by:  wfrlem14  7958  wfr2a  7962
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