Theorem wfrlem15 7429

Description: Lemma for well-founded recursion. When 𝑧 is 𝑅 minimal, 𝐶 is an acceptable function. This step is where the Axiom of Replacement becomes required. (Contributed by Scott Fenton, 21-Apr-2011.)

Hypotheses

Ref	Expression
wfrlem13.1	⊢ 𝑅 We 𝐴
wfrlem13.2	⊢ 𝑅 Se 𝐴
wfrlem13.3	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
wfrlem13.4	⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})

Assertion

Ref	Expression
wfrlem15	⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})

Distinct variable groups:   𝐴,𝑓,𝑥,𝑦,𝑧   𝑓,𝐹,𝑥,𝑦,𝑧   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦,𝑧   𝐶,𝑓,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑧)   𝐺(𝑧)

Proof of Theorem wfrlem15

Step	Hyp	Ref	Expression
1		wfrlem13.1	. . . . . 6 ⊢ 𝑅 We 𝐴
2		wfrlem13.3	. . . . . 6 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
3	1, 2	wfrlem10 7424	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) = dom 𝐹)
4		eldifi 3732	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧 ∈ 𝐴)
5		wfrlem13.2	. . . . . . 7 ⊢ 𝑅 Se 𝐴
6		setlikespec 5701	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
7	4, 5, 6	sylancl 694	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
8	7	adantr 481	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
9	3, 8	eqeltrrd 2702	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → dom 𝐹 ∈ V)
10		snex 4908	. . . 4 ⊢ {𝑧} ∈ V
11		unexg 6959	. . . 4 ⊢ ((dom 𝐹 ∈ V ∧ {𝑧} ∈ V) → (dom 𝐹 ∪ {𝑧}) ∈ V)
12	9, 10, 11	sylancl 694	. . 3 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (dom 𝐹 ∪ {𝑧}) ∈ V)
13		wfrlem13.4	. . . . . 6 ⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
14	1, 5, 2, 13	wfrlem13 7427	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
15	14	adantr 481	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
16	2	wfrdmss 7421	. . . . . . 7 ⊢ dom 𝐹 ⊆ 𝐴
17	4	snssd 4340	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → {𝑧} ⊆ 𝐴)
18		unss 3787	. . . . . . . 8 ⊢ ((dom 𝐹 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴) ↔ (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
19	18	biimpi 206	. . . . . . 7 ⊢ ((dom 𝐹 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
20	16, 17, 19	sylancr 695	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
21	20	adantr 481	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
22		elun 3753	. . . . . . . 8 ⊢ (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 ∈ {𝑧}))
23		velsn 4193	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
24	23	orbi2i 541	. . . . . . . 8 ⊢ ((𝑦 ∈ dom 𝐹 ∨ 𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧))
25	22, 24	bitri 264	. . . . . . 7 ⊢ (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧))
26	2	wfrdmcl 7423	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹)
27		ssun3 3778	. . . . . . . . . 10 ⊢ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}))
28	26, 27	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}))
29	28	a1i 11	. . . . . . . 8 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
30		ssun1 3776	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ (dom 𝐹 ∪ {𝑧})
31	3, 30	syl6eqss 3655	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) ⊆ (dom 𝐹 ∪ {𝑧}))
32		predeq3 5684	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
33	32	sseq1d 3632	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}) ↔ Pred(𝑅, 𝐴, 𝑧) ⊆ (dom 𝐹 ∪ {𝑧})))
34	31, 33	syl5ibrcom 237	. . . . . . . 8 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
35	29, 34	jaod 395	. . . . . . 7 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ((𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧) → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
36	25, 35	syl5bi 232	. . . . . 6 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
37	36	ralrimiv 2965	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}))
38	21, 37	jca 554	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ((dom 𝐹 ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
39	1, 5, 2, 13	wfrlem14 7428	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
40	39	ralrimiv 2965	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))
41	40	adantr 481	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))
42	15, 38, 41	3jca 1242	. . 3 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ ((dom 𝐹 ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})) ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
43		fneq2 5980	. . . . 5 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → (𝐶 Fn 𝑥 ↔ 𝐶 Fn (dom 𝐹 ∪ {𝑧})))
44		sseq1 3626	. . . . . 6 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → (𝑥 ⊆ 𝐴 ↔ (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴))
45		sseq2 3627	. . . . . . 7 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
46	45	raleqbi1dv 3146	. . . . . 6 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
47	44, 46	anbi12d 747	. . . . 5 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → ((𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ↔ ((dom 𝐹 ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}))))
48		raleq 3138	. . . . 5 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → (∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
49	43, 47, 48	3anbi123d 1399	. . . 4 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → ((𝐶 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ ((dom 𝐹 ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})) ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
50	49	spcegv 3294	. . 3 ⊢ ((dom 𝐹 ∪ {𝑧}) ∈ V → ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ ((dom 𝐹 ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})) ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))) → ∃𝑥(𝐶 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
51	12, 42, 50	sylc 65	. 2 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ∃𝑥(𝐶 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
52	10, 11	mpan2 707	. . . . 5 ⊢ (dom 𝐹 ∈ V → (dom 𝐹 ∪ {𝑧}) ∈ V)
53		fnex 6481	. . . . 5 ⊢ ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ (dom 𝐹 ∪ {𝑧}) ∈ V) → 𝐶 ∈ V)
54	52, 53	sylan2 491	. . . 4 ⊢ ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ dom 𝐹 ∈ V) → 𝐶 ∈ V)
55	15, 9, 54	syl2anc 693	. . 3 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ∈ V)
56		fneq1 5979	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝑓 Fn 𝑥 ↔ 𝐶 Fn 𝑥))
57		fveq1 6190	. . . . . . . 8 ⊢ (𝑓 = 𝐶 → (𝑓‘𝑦) = (𝐶‘𝑦))
58		reseq1 5390	. . . . . . . . 9 ⊢ (𝑓 = 𝐶 → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))
59	58	fveq2d 6195	. . . . . . . 8 ⊢ (𝑓 = 𝐶 → (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))
60	57, 59	eqeq12d 2637	. . . . . . 7 ⊢ (𝑓 = 𝐶 → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
61	60	ralbidv 2986	. . . . . 6 ⊢ (𝑓 = 𝐶 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
62	56, 61	3anbi13d 1401	. . . . 5 ⊢ (𝑓 = 𝐶 → ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝐶 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
63	62	exbidv 1850	. . . 4 ⊢ (𝑓 = 𝐶 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑥(𝐶 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
64	63	elabg 3351	. . 3 ⊢ (𝐶 ∈ V → (𝐶 ∈ {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(𝐶 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
65	55, 64	syl 17	. 2 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝐶 ∈ {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(𝐶 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
66	51, 65	mpbird 247	1 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183   Se wse 5071   We wwe 5072  dom cdm 5114   ↾ cres 5116  Predcpred 5679   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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-wrecs 7407

This theorem is referenced by:  wfrlem16  7430