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Theorem wfrlem12 7371
 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 3189 . . 3 𝑦 ∈ V
21eldm2 5282 . 2 (𝑦 ∈ dom 𝐹 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐹)
3 wfrfun.3 . . . . . . 7 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4 df-wrecs 7352 . . . . . . 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 2643 . . . . . 6 𝐹 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
65eleq2i 2690 . . . . 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 4413 . . . . 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 264 . . . 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 2609 . . . . . . . 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 4433 . . . . . . . . 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, 5syl6sseqr 3631 . . . . . . . 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 225 . . . . . . 7 (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → 𝑓𝐹)
13 fnop 5952 . . . . . . . . . . 11 ((𝑓 Fn 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) → 𝑦𝑥)
1413ex 450 . . . . . . . . . 10 (𝑓 Fn 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝑓𝑦𝑥))
15 rsp 2924 . . . . . . . . . . . . . . 15 (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑦𝑥 → (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
1615impcom 446 . . . . . . . . . . . . . 14 ((𝑦𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
17 rsp 2924 . . . . . . . . . . . . . . . . 17 (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦𝑥 → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
18 fndm 5948 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
1918sseq2d 3612 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 Fn 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
2018eleq2d 2684 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 Fn 𝑥 → (𝑦 ∈ dom 𝑓𝑦𝑥))
2119, 20anbi12d 746 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 Fn 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥𝑦𝑥)))
2221biimprd 238 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 Fn 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥𝑦𝑥) → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)))
2322expd 452 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 Fn 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓))))
2423impcom 446 . . . . . . . . . . . . . . . . . . . 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 7370 . . . . . . . . . . . . . . . . . . . . . . 23 Fun 𝐹
28 funssfv 6166 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Fun 𝐹𝑓𝐹𝑦 ∈ dom 𝑓) → (𝐹𝑦) = (𝑓𝑦))
29283adant3l 1319 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Fun 𝐹𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → (𝐹𝑦) = (𝑓𝑦))
30 fun2ssres 5889 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Fun 𝐹𝑓𝐹 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
31303adant3r 1320 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Fun 𝐹𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
3231fveq2d 6152 . . . . . . . . . . . . . . . . . . . . . . . . 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 2636 . . . . . . . . . . . . . . . . . . . . . . . 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 238 . . . . . . . . . . . . . . . . . . . . . . 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 1408 . . . . . . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . 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 1278 . . . . . . . 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 1858 . . . . . . 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 445 . . . . 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 1855 . . . 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 207 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
5554exlimiv 1855 . 2 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
562, 55sylbi 207 1 (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480  ∃wex 1701   ∈ wcel 1987  {cab 2607  ∀wral 2907   ⊆ wss 3555  ⟨cop 4154  ∪ cuni 4402   Se wse 5031   We wwe 5032  dom cdm 5074   ↾ cres 5076  Predcpred 5638  Fun wfun 5841   Fn wfn 5842  ‘cfv 5847  wrecscwrecs 7351 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-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867 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-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-wrecs 7352 This theorem is referenced by:  wfrlem14  7373  wfr2a  7377
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