Theorem wfrlem12 7426

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 → (𝐹‘𝑦) = (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.

Step	Hyp	Ref	Expression
1		vex 3203	. . 3 ⊢ 𝑦 ∈ V
2	1	eldm2 5322	. 2 ⊢ (𝑦 ∈ dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐹)
3		wfrfun.3	. . . . . . 7 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4		df-wrecs 7407	. . . . . . 7 ⊢ wrecs(𝑅, 𝐴, 𝐺) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
5	3, 4	eqtri 2644	. . . . . 6 ⊢ 𝐹 = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
6	5	eleq2i 2693	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐹 ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
7		eluniab 4447	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))))
8	6, 7	bitri 264	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐹 ↔ ∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))))
9		abid 2610	. . . . . . . 8 ⊢ (𝑓 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
10		elssuni 4467	. . . . . . . . 9 ⊢ (𝑓 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} → 𝑓 ⊆ ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})
11	10, 5	syl6sseqr 3652	. . . . . . . 8 ⊢ (𝑓 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} → 𝑓 ⊆ 𝐹)
12	9, 11	sylbir 225	. . . . . . 7 ⊢ (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → 𝑓 ⊆ 𝐹)
13		fnop 5994	. . . . . . . . . . 11 ⊢ ((𝑓 Fn 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) → 𝑦 ∈ 𝑥)
14	13	ex 450	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝑓 → 𝑦 ∈ 𝑥))
15		rsp 2929	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑦 ∈ 𝑥 → (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
16	15	impcom 446	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
17		rsp 2929	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦 ∈ 𝑥 → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
18		fndm 5990	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
19	18	sseq2d 3633	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 Fn 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
20	18	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 Fn 𝑥 → (𝑦 ∈ dom 𝑓 ↔ 𝑦 ∈ 𝑥))
21	19, 20	anbi12d 747	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 Fn 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑥)))
22	21	biimprd 238	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 Fn 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑥) → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓)))
23	22	expd 452	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 Fn 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦 ∈ 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓))))
24	23	impcom 446	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ 𝑓 Fn 𝑥) → (𝑦 ∈ 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓)))
25		wfrfun.1	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑅 We 𝐴
26		wfrfun.2	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑅 Se 𝐴
27	25, 26, 3	wfrfun 7425	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Fun 𝐹
28		funssfv 6209	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ 𝑦 ∈ dom 𝑓) → (𝐹‘𝑦) = (𝑓‘𝑦))
29	28	3adant3l 1322	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓)) → (𝐹‘𝑦) = (𝑓‘𝑦))
30		fun2ssres 5931	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
31	30	3adant3r 1323	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓)) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
32	31	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓)) → (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
33	29, 32	eqeq12d 2637	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓)) → ((𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
34	33	biimprd 238	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓)) → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
35	27, 34	mp3an1 1411	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ⊆ 𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓)) → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
36	35	expcom 451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓) → (𝑓 ⊆ 𝐹 → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
37	36	com23 86	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓) → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓 ⊆ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
38	24, 37	syl6com 37	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ 𝑓 Fn 𝑥) → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓 ⊆ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))))
39	38	expd 452	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑓 Fn 𝑥 → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓 ⊆ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
40	39	com34 91	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓 Fn 𝑥 → (𝑓 ⊆ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
41	17, 40	sylcom 30	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦 ∈ 𝑥 → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓 Fn 𝑥 → (𝑓 ⊆ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
42	41	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑦 ∈ 𝑥 → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓 Fn 𝑥 → (𝑓 ⊆ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
43	42	com14 96	. . . . . . . . . . . . . 14 ⊢ (𝑓 Fn 𝑥 → (𝑦 ∈ 𝑥 → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → ((𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑓 ⊆ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
44	16, 43	syl7 74	. . . . . . . . . . . . 13 ⊢ (𝑓 Fn 𝑥 → (𝑦 ∈ 𝑥 → ((𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → ((𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑓 ⊆ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
45	44	exp4a 633	. . . . . . . . . . . 12 ⊢ (𝑓 Fn 𝑥 → (𝑦 ∈ 𝑥 → (𝑦 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → ((𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑓 ⊆ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))))))
46	45	pm2.43d 53	. . . . . . . . . . 11 ⊢ (𝑓 Fn 𝑥 → (𝑦 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → ((𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (𝑓 ⊆ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
47	46	com34 91	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑥 → (𝑦 ∈ 𝑥 → ((𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓 ⊆ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
48	14, 47	syldc 48	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝑓 → (𝑓 Fn 𝑥 → ((𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑓 ⊆ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))))
49	48	3impd 1281	. . . . . . . 8 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝑓 → ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝑓 ⊆ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
50	49	exlimdv 1861	. . . . . . 7 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝑓 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝑓 ⊆ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))))
51	12, 50	mpdi 45	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝑓 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))))
52	51	imp 445	. . . . 5 ⊢ ((⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
53	52	exlimiv 1858	. . . 4 ⊢ (∃𝑓(⟨𝑦, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
54	8, 53	sylbi 207	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
55	54	exlimiv 1858	. 2 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
56	2, 55	sylbi 207	1 ⊢ (𝑦 ∈ dom 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912   ⊆ wss 3574  ⟨cop 4183  ∪ cuni 4436   Se wse 5071   We wwe 5072  dom cdm 5114   ↾ cres 5116  Predcpred 5679  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888  wrecscwrecs 7406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-wrecs 7407

This theorem is referenced by:  wfrlem14  7428  wfr2a  7432