Theorem tfrcl 6368

Description: Closure for transfinite recursion. As with tfr1on 6354, the characteristic function must be defined up to a suitable point, not necessarily on all ordinals. (Contributed by Jim Kingdon, 25-Mar-2022.)

Hypotheses

Ref	Expression
tfrcl.f	⊢ 𝐹 = recs(𝐺)
tfrcl.g	⊢ (𝜑 → Fun 𝐺)
tfrcl.x	⊢ (𝜑 → Ord 𝑋)
tfrcl.ex	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
tfrcl.u	⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
tfrcl.yx	⊢ (𝜑 → 𝑌 ∈ ∪ 𝑋)

Assertion

Ref	Expression
tfrcl	⊢ (𝜑 → (𝐹‘𝑌) ∈ 𝑆)

Distinct variable groups:   𝑓,𝐹,𝑥   𝑓,𝐺,𝑥   𝑆,𝑓,𝑥   𝑓,𝑋,𝑥   𝜑,𝑓,𝑥

Allowed substitution hints:   𝑌(𝑥,𝑓)

Proof of Theorem tfrcl

Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrcl.x	. . . 4 ⊢ (𝜑 → Ord 𝑋)
2		orduni 4496	. . . 4 ⊢ (Ord 𝑋 → Ord ∪ 𝑋)
3	1, 2	syl 14	. . 3 ⊢ (𝜑 → Ord ∪ 𝑋)
4		tfrcl.yx	. . 3 ⊢ (𝜑 → 𝑌 ∈ ∪ 𝑋)
5		ordelon 4385	. . 3 ⊢ ((Ord ∪ 𝑋 ∧ 𝑌 ∈ ∪ 𝑋) → 𝑌 ∈ On)
6	3, 4, 5	syl2anc 411	. 2 ⊢ (𝜑 → 𝑌 ∈ On)
7	4	ancli 323	. 2 ⊢ (𝜑 → (𝜑 ∧ 𝑌 ∈ ∪ 𝑋))
8		eleq1 2240	. . . . 5 ⊢ (𝑤 = 𝑘 → (𝑤 ∈ ∪ 𝑋 ↔ 𝑘 ∈ ∪ 𝑋))
9	8	anbi2d 464	. . . 4 ⊢ (𝑤 = 𝑘 → ((𝜑 ∧ 𝑤 ∈ ∪ 𝑋) ↔ (𝜑 ∧ 𝑘 ∈ ∪ 𝑋)))
10		fveq2 5517	. . . . 5 ⊢ (𝑤 = 𝑘 → (𝐹‘𝑤) = (𝐹‘𝑘))
11	10	eleq1d 2246	. . . 4 ⊢ (𝑤 = 𝑘 → ((𝐹‘𝑤) ∈ 𝑆 ↔ (𝐹‘𝑘) ∈ 𝑆))
12	9, 11	imbi12d 234	. . 3 ⊢ (𝑤 = 𝑘 → (((𝜑 ∧ 𝑤 ∈ ∪ 𝑋) → (𝐹‘𝑤) ∈ 𝑆) ↔ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)))
13		eleq1 2240	. . . . 5 ⊢ (𝑤 = 𝑌 → (𝑤 ∈ ∪ 𝑋 ↔ 𝑌 ∈ ∪ 𝑋))
14	13	anbi2d 464	. . . 4 ⊢ (𝑤 = 𝑌 → ((𝜑 ∧ 𝑤 ∈ ∪ 𝑋) ↔ (𝜑 ∧ 𝑌 ∈ ∪ 𝑋)))
15		fveq2 5517	. . . . 5 ⊢ (𝑤 = 𝑌 → (𝐹‘𝑤) = (𝐹‘𝑌))
16	15	eleq1d 2246	. . . 4 ⊢ (𝑤 = 𝑌 → ((𝐹‘𝑤) ∈ 𝑆 ↔ (𝐹‘𝑌) ∈ 𝑆))
17	14, 16	imbi12d 234	. . 3 ⊢ (𝑤 = 𝑌 → (((𝜑 ∧ 𝑤 ∈ ∪ 𝑋) → (𝐹‘𝑤) ∈ 𝑆) ↔ ((𝜑 ∧ 𝑌 ∈ ∪ 𝑋) → (𝐹‘𝑌) ∈ 𝑆)))
18		tfrcl.f	. . . . . . 7 ⊢ 𝐹 = recs(𝐺)
19		tfrcl.g	. . . . . . . 8 ⊢ (𝜑 → Fun 𝐺)
20	19	ad2antrl 490	. . . . . . 7 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → Fun 𝐺)
21	1	ad2antrl 490	. . . . . . 7 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → Ord 𝑋)
22		tfrcl.ex	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
23	22	3adant1r 1231	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ ∪ 𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
24	23	3adant1l 1230	. . . . . . 7 ⊢ ((((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
25		tfrcl.u	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
26	25	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ ∪ 𝑋) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
27	26	adantll 476	. . . . . . 7 ⊢ ((((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
28		simprr 531	. . . . . . 7 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → 𝑤 ∈ ∪ 𝑋)
29	18, 20, 21, 24, 27, 28	tfrcldm 6367	. . . . . 6 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → 𝑤 ∈ dom 𝐹)
30	18	tfr2a 6325	. . . . . 6 ⊢ (𝑤 ∈ dom 𝐹 → (𝐹‘𝑤) = (𝐺‘(𝐹 ↾ 𝑤)))
31	29, 30	syl 14	. . . . 5 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → (𝐹‘𝑤) = (𝐺‘(𝐹 ↾ 𝑤)))
32	19	ad2antrl 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → Fun 𝐺)
33	32	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) → Fun 𝐺)
34	33	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) ∧ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) → Fun 𝐺)
35	1	ad2antrl 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → Ord 𝑋)
36	35	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) → Ord 𝑋)
37	36	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) ∧ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) → Ord 𝑋)
38		simplrl 535	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) → 𝜑)
39	38, 22	syl3an1 1271	. . . . . . . . . . . . . . 15 ⊢ ((((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
40	39	3adant1r 1231	. . . . . . . . . . . . . 14 ⊢ (((((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) ∧ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
41	38, 25	sylan 283	. . . . . . . . . . . . . . 15 ⊢ ((((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
42	41	adantlr 477	. . . . . . . . . . . . . 14 ⊢ (((((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) ∧ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
43	36, 2	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) → Ord ∪ 𝑋)
44		simpr 110	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) → 𝑘 ∈ 𝑤)
45		simplrr 536	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) → 𝑤 ∈ ∪ 𝑋)
46	44, 45	jca 306	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) → (𝑘 ∈ 𝑤 ∧ 𝑤 ∈ ∪ 𝑋))
47		ordtr1 4390	. . . . . . . . . . . . . . . 16 ⊢ (Ord ∪ 𝑋 → ((𝑘 ∈ 𝑤 ∧ 𝑤 ∈ ∪ 𝑋) → 𝑘 ∈ ∪ 𝑋))
48	43, 46, 47	sylc 62	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) → 𝑘 ∈ ∪ 𝑋)
49	48	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) ∧ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) → 𝑘 ∈ ∪ 𝑋)
50	18, 34, 37, 40, 42, 49	tfrcldm 6367	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) ∧ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) → 𝑘 ∈ dom 𝐹)
51	38, 48	jca 306	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) → (𝜑 ∧ 𝑘 ∈ ∪ 𝑋))
52	51	imim1i 60	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆) → (((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) → (𝐹‘𝑘) ∈ 𝑆))
53	52	impcom 125	. . . . . . . . . . . . 13 ⊢ ((((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) ∧ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) → (𝐹‘𝑘) ∈ 𝑆)
54	50, 53	jca 306	. . . . . . . . . . . 12 ⊢ ((((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) ∧ ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑆))
55	54	ex 115	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ 𝑘 ∈ 𝑤) → (((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆) → (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑆)))
56	55	ralimdva 2544	. . . . . . . . . 10 ⊢ ((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → (∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆) → ∀𝑘 ∈ 𝑤 (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑆)))
57	56	imp 124	. . . . . . . . 9 ⊢ (((𝑤 ∈ On ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) → ∀𝑘 ∈ 𝑤 (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑆))
58	57	an32s 568	. . . . . . . 8 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → ∀𝑘 ∈ 𝑤 (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑆))
59		tfrfun 6324	. . . . . . . . . . 11 ⊢ Fun recs(𝐺)
60	18	funeqi 5239	. . . . . . . . . . 11 ⊢ (Fun 𝐹 ↔ Fun recs(𝐺))
61	59, 60	mpbir 146	. . . . . . . . . 10 ⊢ Fun 𝐹
62	61	a1i 9	. . . . . . . . 9 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → Fun 𝐹)
63		ffvresb 5682	. . . . . . . . 9 ⊢ (Fun 𝐹 → ((𝐹 ↾ 𝑤):𝑤⟶𝑆 ↔ ∀𝑘 ∈ 𝑤 (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑆)))
64	62, 63	syl 14	. . . . . . . 8 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → ((𝐹 ↾ 𝑤):𝑤⟶𝑆 ↔ ∀𝑘 ∈ 𝑤 (𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑆)))
65	58, 64	mpbird 167	. . . . . . 7 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → (𝐹 ↾ 𝑤):𝑤⟶𝑆)
66		vex 2742	. . . . . . 7 ⊢ 𝑤 ∈ V
67		fex 5748	. . . . . . 7 ⊢ (((𝐹 ↾ 𝑤):𝑤⟶𝑆 ∧ 𝑤 ∈ V) → (𝐹 ↾ 𝑤) ∈ V)
68	65, 66, 67	sylancl 413	. . . . . 6 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → (𝐹 ↾ 𝑤) ∈ V)
69		feq2 5351	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑓:𝑥⟶𝑆 ↔ 𝑓:𝑤⟶𝑆))
70	69	imbi1d 231	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (𝑓:𝑤⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)))
71	70	albidv 1824	. . . . . . 7 ⊢ (𝑥 = 𝑤 → (∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ ∀𝑓(𝑓:𝑤⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)))
72	22	3expia 1205	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
73	72	alrimiv 1874	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
74	73	ralrimiva 2550	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
75	74	ad2antrl 490	. . . . . . 7 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
76	66	sucid 4419	. . . . . . . . . 10 ⊢ 𝑤 ∈ suc 𝑤
77	76	a1i 9	. . . . . . . . 9 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → 𝑤 ∈ suc 𝑤)
78		suceq 4404	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → suc 𝑥 = suc 𝑤)
79	78	eleq1d 2246	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → (suc 𝑥 ∈ 𝑋 ↔ suc 𝑤 ∈ 𝑋))
80	25	ralrimiva 2550	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ ∪ 𝑋 suc 𝑥 ∈ 𝑋)
81	80	ad2antrl 490	. . . . . . . . . 10 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → ∀𝑥 ∈ ∪ 𝑋 suc 𝑥 ∈ 𝑋)
82	79, 81, 28	rspcdva 2848	. . . . . . . . 9 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → suc 𝑤 ∈ 𝑋)
83	77, 82	jca 306	. . . . . . . 8 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → (𝑤 ∈ suc 𝑤 ∧ suc 𝑤 ∈ 𝑋))
84		ordtr1 4390	. . . . . . . 8 ⊢ (Ord 𝑋 → ((𝑤 ∈ suc 𝑤 ∧ suc 𝑤 ∈ 𝑋) → 𝑤 ∈ 𝑋))
85	21, 83, 84	sylc 62	. . . . . . 7 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → 𝑤 ∈ 𝑋)
86	71, 75, 85	rspcdva 2848	. . . . . 6 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → ∀𝑓(𝑓:𝑤⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
87		feq1 5350	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ 𝑤) → (𝑓:𝑤⟶𝑆 ↔ (𝐹 ↾ 𝑤):𝑤⟶𝑆))
88		fveq2 5517	. . . . . . . . 9 ⊢ (𝑓 = (𝐹 ↾ 𝑤) → (𝐺‘𝑓) = (𝐺‘(𝐹 ↾ 𝑤)))
89	88	eleq1d 2246	. . . . . . . 8 ⊢ (𝑓 = (𝐹 ↾ 𝑤) → ((𝐺‘𝑓) ∈ 𝑆 ↔ (𝐺‘(𝐹 ↾ 𝑤)) ∈ 𝑆))
90	87, 89	imbi12d 234	. . . . . . 7 ⊢ (𝑓 = (𝐹 ↾ 𝑤) → ((𝑓:𝑤⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ ((𝐹 ↾ 𝑤):𝑤⟶𝑆 → (𝐺‘(𝐹 ↾ 𝑤)) ∈ 𝑆)))
91	90	spcgv 2826	. . . . . 6 ⊢ ((𝐹 ↾ 𝑤) ∈ V → (∀𝑓(𝑓:𝑤⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) → ((𝐹 ↾ 𝑤):𝑤⟶𝑆 → (𝐺‘(𝐹 ↾ 𝑤)) ∈ 𝑆)))
92	68, 86, 65, 91	syl3c 63	. . . . 5 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → (𝐺‘(𝐹 ↾ 𝑤)) ∈ 𝑆)
93	31, 92	eqeltrd 2254	. . . 4 ⊢ (((𝑤 ∈ On ∧ ∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆)) ∧ (𝜑 ∧ 𝑤 ∈ ∪ 𝑋)) → (𝐹‘𝑤) ∈ 𝑆)
94	93	exp31 364	. . 3 ⊢ (𝑤 ∈ On → (∀𝑘 ∈ 𝑤 ((𝜑 ∧ 𝑘 ∈ ∪ 𝑋) → (𝐹‘𝑘) ∈ 𝑆) → ((𝜑 ∧ 𝑤 ∈ ∪ 𝑋) → (𝐹‘𝑤) ∈ 𝑆)))
95	12, 17, 94	tfis3 4587	. 2 ⊢ (𝑌 ∈ On → ((𝜑 ∧ 𝑌 ∈ ∪ 𝑋) → (𝐹‘𝑌) ∈ 𝑆))
96	6, 7, 95	sylc 62	1 ⊢ (𝜑 → (𝐹‘𝑌) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978  ∀wal 1351   = wceq 1353   ∈ wcel 2148  ∀wral 2455  Vcvv 2739  ∪ cuni 3811  Ord word 4364  Oncon0 4365  suc csuc 4367  dom cdm 4628   ↾ cres 4630  Fun wfun 5212  ⟶wf 5214  ‘cfv 5218  recscrecs 6308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-recs 6309

This theorem is referenced by:  rdgon  6390  freccllem  6406  frecfcllem  6408