Theorem wfrlem12OLD 8376

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 → (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 wfrlem12OLD

Dummy variables 𝑓 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3492	. . 3 ⊢ 𝑦 ∈ V
2	1	eldm2 5926	. 2 ⊢ (𝑦 ∈ dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐹)
3		wfrfunOLD.3	. . . . . . 7 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4		dfwrecsOLD 8354	. . . . . . 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 2768	. . . . . 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 2836	. . . . 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 4945	. . . . 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 275	. . . 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 2721	. . . . . . . 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 4961	. . . . . . . . 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	sseqtrrdi 4060	. . . . . . . 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 235	. . . . . . 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 6688	. . . . . . . . . . 11 ⊢ ((𝑓 Fn 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) → 𝑦 ∈ 𝑥)
14	13	ex 412	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝑓 → 𝑦 ∈ 𝑥))
15		rsp 3253	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑦 ∈ 𝑥 → (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
16	15	impcom 407	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
17		rsp 3253	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦 ∈ 𝑥 → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
18		fndm 6682	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
19	18	sseq2d 4041	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 Fn 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
20	18	eleq2d 2830	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 Fn 𝑥 → (𝑦 ∈ dom 𝑓 ↔ 𝑦 ∈ 𝑥))
21	19, 20	anbi12d 631	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 Fn 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑥)))
22	21	biimprd 248	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 Fn 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑥) → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓)))
23	22	expd 415	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 Fn 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦 ∈ 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓))))
24	23	impcom 407	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ 𝑓 Fn 𝑥) → (𝑦 ∈ 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓)))
25		wfrfunOLD.1	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑅 We 𝐴
26		wfrfunOLD.2	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑅 Se 𝐴
27	25, 26, 3	wfrfunOLD 8375	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Fun 𝐹
28		funssfv 6941	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ 𝑦 ∈ dom 𝑓) → (𝐹‘𝑦) = (𝑓‘𝑦))
29	28	3adant3l 1180	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓)) → (𝐹‘𝑦) = (𝑓‘𝑦))
30		fun2ssres 6623	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
31	30	3adant3r 1181	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6924	. . . . . . . . . . . . . . . . . . . . . . . . 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 2756	. . . . . . . . . . . . . . . . . . . . . . . 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 248	. . . . . . . . . . . . . . . . . . . . . . 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 1448	. . . . . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . 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 431	. . . . . . . . . . . 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 1348	. . . . . . . 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 1932	. . . . . . 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 406	. . . . 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 1929	. . . 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 217	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
55	54	exlimiv 1929	. 2 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
56	2, 55	sylbi 217	1 ⊢ (𝑦 ∈ dom 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  ∀wral 3067   ⊆ wss 3976  ⟨cop 4654  ∪ cuni 4931   Se wse 5650   We wwe 5651  dom cdm 5700   ↾ cres 5702  Predcpred 6331  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573  wrecscwrecs 8352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-ov 7451  df-2nd 8031  df-frecs 8322  df-wrecs 8353

This theorem is referenced by:  wfrlem14OLD  8378  wfr2aOLD  8382