Theorem tfrcllemres 6514

Description: Lemma for tfr1on 6502. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-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)
tfrcllemsucfn.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
tfrcllemres.u	⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
tfrcllemres.yx	⊢ (𝜑 → 𝑌 ∈ 𝑋)

Assertion

Ref	Expression
tfrcllemres	⊢ (𝜑 → 𝑌 ⊆ dom 𝐹)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑓,𝐺,𝑥,𝑦   𝑆,𝑓,𝑥,𝑦   𝑓,𝑋,𝑥,𝑦   𝑓,𝑌,𝑥,𝑦   𝜑,𝑓,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑓)   𝐹(𝑥,𝑦,𝑓)

Proof of Theorem tfrcllemres

Dummy variables 𝑔 ℎ 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrcl.x	. . . . . . . . . 10 ⊢ (𝜑 → Ord 𝑋)
2	1	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → Ord 𝑋)
3		simpr 110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → 𝑧 ∈ 𝑌)
4		tfrcllemres.yx	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
5	4	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → 𝑌 ∈ 𝑋)
6	3, 5	jca 306	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → (𝑧 ∈ 𝑌 ∧ 𝑌 ∈ 𝑋))
7		ordtr1 4479	. . . . . . . . 9 ⊢ (Ord 𝑋 → ((𝑧 ∈ 𝑌 ∧ 𝑌 ∈ 𝑋) → 𝑧 ∈ 𝑋))
8	2, 6, 7	sylc 62	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → 𝑧 ∈ 𝑋)
9		tfrcl.f	. . . . . . . . 9 ⊢ 𝐹 = recs(𝐺)
10		tfrcl.g	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐺)
11		tfrcl.ex	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
12		tfrcllemsucfn.1	. . . . . . . . 9 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
13		tfrcllemres.u	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
14	9, 10, 1, 11, 12, 13	tfrcllemaccex 6513	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
15	8, 14	syldan 282	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
16	10	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → Fun 𝐺)
17	1	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → Ord 𝑋)
18	11	3adant1r 1255	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
19	18	3adant1r 1255	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
20	4	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → 𝑌 ∈ 𝑋)
21	3	adantr 276	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → 𝑧 ∈ 𝑌)
22	13	adantlr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
23	22	adantlr 477	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
24		simprl 529	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → 𝑔:𝑧⟶𝑆)
25		feq2 5457	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑔:𝑥⟶𝑆 ↔ 𝑔:𝑧⟶𝑆))
26		raleq 2728	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
27	25, 26	anbi12d 473	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))) ↔ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))))
28		fveq2 5629	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → (𝑔‘𝑦) = (𝑔‘𝑢))
29		reseq2 5000	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑢 → (𝑔 ↾ 𝑦) = (𝑔 ↾ 𝑢))
30	29	fveq2d 5633	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → (𝐺‘(𝑔 ↾ 𝑦)) = (𝐺‘(𝑔 ↾ 𝑢)))
31	28, 30	eqeq12d 2244	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑢 → ((𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)) ↔ (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
32	31	cbvralv 2765	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)) ↔ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))
33	32	anbi2i 457	. . . . . . . . . . . . 13 ⊢ ((𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))) ↔ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
34	27, 33	bitrdi 196	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))) ↔ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))
35	34	rspcev 2907	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ∃𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
36	8, 35	sylan 283	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ∃𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
37		vex 2802	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
38		feq1 5456	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝑓:𝑥⟶𝑆 ↔ 𝑔:𝑥⟶𝑆))
39		fveq1 5628	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑦) = (𝑔‘𝑦))
40		reseq1 4999	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ 𝑦) = (𝑔 ↾ 𝑦))
41	40	fveq2d 5633	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘(𝑔 ↾ 𝑦)))
42	39, 41	eqeq12d 2244	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
43	42	ralbidv 2530	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
44	38, 43	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))))
45	44	rexbidv 2531	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ ∃𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))))
46	37, 45, 12	elab2 2951	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
47	36, 46	sylibr 134	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → 𝑔 ∈ 𝐴)
48	9, 16, 17, 19, 12, 20, 21, 23, 24, 47	tfrcllemsucaccv 6506	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴)
49		vex 2802	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
50	25	imbi1d 231	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝑔:𝑥⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆) ↔ (𝑔:𝑧⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆)))
51	11	3expia 1229	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
52	51	alrimiv 1920	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
53		fveq2 5629	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔))
54	53	eleq1d 2298	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ 𝑆 ↔ (𝐺‘𝑔) ∈ 𝑆))
55	38, 54	imbi12d 234	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = 𝑔 → ((𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (𝑔:𝑥⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆)))
56	55	spv 1906	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) → (𝑔:𝑥⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆))
57	52, 56	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑔:𝑥⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆))
58	57	ralrimiva 2603	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆))
59	58	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆))
60	50, 59, 8	rspcdva 2912	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → (𝑔:𝑧⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆))
61	60	imp 124	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ 𝑔:𝑧⟶𝑆) → (𝐺‘𝑔) ∈ 𝑆)
62	24, 61	syldan 282	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → (𝐺‘𝑔) ∈ 𝑆)
63		opexg 4314	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ 𝑆) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
64	49, 62, 63	sylancr 414	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
65		snidg 3695	. . . . . . . . . 10 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ V → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ {⟨𝑧, (𝐺‘𝑔)⟩})
66		elun2 3372	. . . . . . . . . 10 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ {⟨𝑧, (𝐺‘𝑔)⟩} → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
67	64, 65, 66	3syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
68		opeldmg 4928	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ 𝑆) → (⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
69	49, 62, 68	sylancr 414	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → (⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
70	67, 69	mpd 13	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
71		dmeq 4923	. . . . . . . . . 10 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → dom ℎ = dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
72	71	eleq2d 2299	. . . . . . . . 9 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑧 ∈ dom ℎ ↔ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
73	72	rspcev 2907	. . . . . . . 8 ⊢ (((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴 ∧ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
74	48, 70, 73	syl2anc 411	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
75	15, 74	exlimddv 1945	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
76		eliun 3969	. . . . . 6 ⊢ (𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ ↔ ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
77	75, 76	sylibr 134	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → 𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ)
78	77	ex 115	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝑌 → 𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ))
79	78	ssrdv 3230	. . 3 ⊢ (𝜑 → 𝑌 ⊆ ∪ ℎ ∈ 𝐴 dom ℎ)
80		dmuni 4933	. . . 4 ⊢ dom ∪ 𝐴 = ∪ ℎ ∈ 𝐴 dom ℎ
81	12, 1	tfrcllemssrecs 6504	. . . . 5 ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺))
82		dmss 4922	. . . . 5 ⊢ (∪ 𝐴 ⊆ recs(𝐺) → dom ∪ 𝐴 ⊆ dom recs(𝐺))
83	81, 82	syl 14	. . . 4 ⊢ (𝜑 → dom ∪ 𝐴 ⊆ dom recs(𝐺))
84	80, 83	eqsstrrid 3271	. . 3 ⊢ (𝜑 → ∪ ℎ ∈ 𝐴 dom ℎ ⊆ dom recs(𝐺))
85	79, 84	sstrd 3234	. 2 ⊢ (𝜑 → 𝑌 ⊆ dom recs(𝐺))
86	9	dmeqi 4924	. 2 ⊢ dom 𝐹 = dom recs(𝐺)
87	85, 86	sseqtrrdi 3273	1 ⊢ (𝜑 → 𝑌 ⊆ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002  ∀wal 1393   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509  Vcvv 2799   ∪ cun 3195   ⊆ wss 3197  {csn 3666  ⟨cop 3669  ∪ cuni 3888  ∪ ciun 3965  Ord word 4453  suc csuc 4456  dom cdm 4719   ↾ cres 4721  Fun wfun 5312  ⟶wf 5314  ‘cfv 5318  recscrecs 6456

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-recs 6457

This theorem is referenced by:  tfrcldm  6515