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Theorem wfrlem12OLD 8267
Description: Lemma for well-ordered recursion. Here, we compute the value of the recursive definition generator. Obsolete as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfrfunOLD.1 𝑅 We 𝐴
wfrfunOLD.2 𝑅 Se 𝐴
wfrfunOLD.3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
wfrlem12OLD (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 wfrlem12OLD
Dummy variables 𝑓 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3448 . . 3 𝑦 ∈ V
21eldm2 5858 . 2 (𝑦 ∈ dom 𝐹 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐹)
3 wfrfunOLD.3 . . . . . . 7 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4 dfwrecsOLD 8245 . . . . . . 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 2761 . . . . . 6 𝐹 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
65eleq2i 2826 . . . . 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 4881 . . . . 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 275 . . . 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 2714 . . . . . . . 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 4899 . . . . . . . . 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 3996 . . . . . . . 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 234 . . . . . . 7 (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → 𝑓𝐹)
13 fnop 6612 . . . . . . . . . . 11 ((𝑓 Fn 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) → 𝑦𝑥)
1413ex 414 . . . . . . . . . 10 (𝑓 Fn 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝑓𝑦𝑥))
15 rsp 3229 . . . . . . . . . . . . . . 15 (∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑦𝑥 → (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
1615impcom 409 . . . . . . . . . . . . . 14 ((𝑦𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝑓𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
17 rsp 3229 . . . . . . . . . . . . . . . . 17 (∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦𝑥 → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
18 fndm 6606 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
1918sseq2d 3977 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 Fn 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
2018eleq2d 2820 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑓 Fn 𝑥 → (𝑦 ∈ dom 𝑓𝑦𝑥))
2119, 20anbi12d 632 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓 Fn 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥𝑦𝑥)))
2221biimprd 248 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓 Fn 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥𝑦𝑥) → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)))
2322expd 417 . . . . . . . . . . . . . . . . . . . . 21 (𝑓 Fn 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓))))
2423impcom 409 . . . . . . . . . . . . . . . . . . . 20 ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥𝑓 Fn 𝑥) → (𝑦𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)))
25 wfrfunOLD.1 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑅 We 𝐴
26 wfrfunOLD.2 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑅 Se 𝐴
2725, 26, 3wfrfunOLD 8266 . . . . . . . . . . . . . . . . . . . . . . 23 Fun 𝐹
28 funssfv 6864 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Fun 𝐹𝑓𝐹𝑦 ∈ dom 𝑓) → (𝐹𝑦) = (𝑓𝑦))
29283adant3l 1181 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((Fun 𝐹𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → (𝐹𝑦) = (𝑓𝑦))
30 fun2ssres 6547 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((Fun 𝐹𝑓𝐹 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
31303adant3r 1182 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((Fun 𝐹𝑓𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓𝑦 ∈ dom 𝑓)) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
3231fveq2d 6847 . . . . . . . . . . . . . . . . . . . . . . . . 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 2749 . . . . . . . . . . . . . . . . . . . . . . . 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 248 . . . . . . . . . . . . . . . . . . . . . . 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 1449 . . . . . . . . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . . . . . . . . . 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 417 . . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . 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 433 . . . . . . . . . . . 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 1349 . . . . . . . 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 1937 . . . . . . 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 408 . . . . 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 1934 . . . 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 216 . . 3 (⟨𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
5554exlimiv 1934 . 2 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
562, 55sylbi 216 1 (𝑦 ∈ dom 𝐹 → (𝐹𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3061  wss 3911  cop 4593   cuni 4866   Se wse 5587   We wwe 5588  dom cdm 5634  cres 5636  Predcpred 6253  Fun wfun 6491   Fn wfn 6492  cfv 6497  wrecscwrecs 8243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-ov 7361  df-2nd 7923  df-frecs 8213  df-wrecs 8244
This theorem is referenced by:  wfrlem14OLD  8269  wfr2aOLD  8273
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