Theorem wfrlem12OLD 8337

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 3467	. . 3 ⊢ 𝑦 ∈ V
2	1	eldm2 5898	. 2 ⊢ (𝑦 ∈ dom 𝐹 ↔ ∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐹)
3		wfrfunOLD.3	. . . . . . 7 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4		dfwrecsOLD 8315	. . . . . . 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 2753	. . . . . 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 2817	. . . . 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 4917	. . . . 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 274	. . . 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 2706	. . . . . . . 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 4935	. . . . . . . . 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 4024	. . . . . . . 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 234	. . . . . . 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 6657	. . . . . . . . . . 11 ⊢ ((𝑓 Fn 𝑥 ∧ ⟨𝑦, 𝑧⟩ ∈ 𝑓) → 𝑦 ∈ 𝑥)
14	13	ex 411	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑥 → (⟨𝑦, 𝑧⟩ ∈ 𝑓 → 𝑦 ∈ 𝑥))
15		rsp 3235	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) → (𝑦 ∈ 𝑥 → (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))
16	15	impcom 406	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))
17		rsp 3235	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦 ∈ 𝑥 → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
18		fndm 6651	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
19	18	sseq2d 4005	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 Fn 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥))
20	18	eleq2d 2811	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 Fn 𝑥 → (𝑦 ∈ dom 𝑓 ↔ 𝑦 ∈ 𝑥))
21	19, 20	anbi12d 630	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 Fn 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓) ↔ (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑥)))
22	21	biimprd 247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 Fn 𝑥 → ((Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ 𝑦 ∈ 𝑥) → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓)))
23	22	expd 414	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 Fn 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 → (𝑦 ∈ 𝑥 → (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓))))
24	23	impcom 406	. . . . . . . . . . . . . . . . . . . 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 8336	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Fun 𝐹
28		funssfv 6912	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ 𝑦 ∈ dom 𝑓) → (𝐹‘𝑦) = (𝑓‘𝑦))
29	28	3adant3l 1177	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓 ∧ 𝑦 ∈ dom 𝑓)) → (𝐹‘𝑦) = (𝑓‘𝑦))
30		fun2ssres 6592	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((Fun 𝐹 ∧ 𝑓 ⊆ 𝐹 ∧ Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝑓) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))
31	30	3adant3r 1178	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6895	. . . . . . . . . . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . . . . . . . . . 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 1444	. . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 430	. . . . . . . . . . . 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 1345	. . . . . . . 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 1928	. . . . . . 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 405	. . . . 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 1925	. . . 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 216	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
55	54	exlimiv 1925	. 2 ⊢ (∃𝑧⟨𝑦, 𝑧⟩ ∈ 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))
56	2, 55	sylbi 216	1 ⊢ (𝑦 ∈ dom 𝐹 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2702  ∀wral 3051   ⊆ wss 3940  ⟨cop 4630  ∪ cuni 4903   Se wse 5625   We wwe 5626  dom cdm 5672   ↾ cres 5674  Predcpred 6299  Fun wfun 6536   Fn wfn 6537  ‘cfv 6542  wrecscwrecs 8313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-ov 7418  df-2nd 7990  df-frecs 8283  df-wrecs 8314

This theorem is referenced by:  wfrlem14OLD  8339  wfr2aOLD  8343