Theorem tfrcllemres 6330

Description: Lemma for tfr1on 6318. 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 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → Ord 𝑋)
3		simpr 109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → 𝑧 ∈ 𝑌)
4		tfrcllemres.yx	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
5	4	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → 𝑌 ∈ 𝑋)
6	3, 5	jca 304	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → (𝑧 ∈ 𝑌 ∧ 𝑌 ∈ 𝑋))
7		ordtr1 4366	. . . . . . . . 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 6329	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
15	8, 14	syldan 280	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
16	10	ad2antrr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → Fun 𝐺)
17	1	ad2antrr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → Ord 𝑋)
18	11	3adant1r 1221	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
19	18	3adant1r 1221	. . . . . . . . 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 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → 𝑌 ∈ 𝑋)
21	3	adantr 274	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → 𝑧 ∈ 𝑌)
22	13	adantlr 469	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
23	22	adantlr 469	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
24		simprl 521	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → 𝑔:𝑧⟶𝑆)
25		feq2 5321	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝑔:𝑥⟶𝑆 ↔ 𝑔:𝑧⟶𝑆))
26		raleq 2661	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
27	25, 26	anbi12d 465	. . . . . . . . . . . . 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 5486	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → (𝑔‘𝑦) = (𝑔‘𝑢))
29		reseq2 4879	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑢 → (𝑔 ↾ 𝑦) = (𝑔 ↾ 𝑢))
30	29	fveq2d 5490	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → (𝐺‘(𝑔 ↾ 𝑦)) = (𝐺‘(𝑔 ↾ 𝑢)))
31	28, 30	eqeq12d 2180	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑢 → ((𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)) ↔ (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
32	31	cbvralv 2692	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)) ↔ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))
33	32	anbi2i 453	. . . . . . . . . . . . 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 195	. . . . . . . . . . . 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 2830	. . . . . . . . . . 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 281	. . . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
38		feq1 5320	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝑓:𝑥⟶𝑆 ↔ 𝑔:𝑥⟶𝑆))
39		fveq1 5485	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑦) = (𝑔‘𝑦))
40		reseq1 4878	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ 𝑦) = (𝑔 ↾ 𝑦))
41	40	fveq2d 5490	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘(𝑔 ↾ 𝑦)))
42	39, 41	eqeq12d 2180	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
43	42	ralbidv 2466	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
44	38, 43	anbi12d 465	. . . . . . . . . . . 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 2467	. . . . . . . . . . 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 2874	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
47	36, 46	sylibr 133	. . . . . . . . 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 6322	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴)
49		vex 2729	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
50	25	imbi1d 230	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝑔:𝑥⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆) ↔ (𝑔:𝑧⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆)))
51	11	3expia 1195	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
52	51	alrimiv 1862	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
53		fveq2 5486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔))
54	53	eleq1d 2235	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ 𝑆 ↔ (𝐺‘𝑔) ∈ 𝑆))
55	38, 54	imbi12d 233	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = 𝑔 → ((𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (𝑔:𝑥⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆)))
56	55	spv 1848	. . . . . . . . . . . . . . . . 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 2539	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆))
59	58	adantr 274	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆))
60	50, 59, 8	rspcdva 2835	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → (𝑔:𝑧⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆))
61	60	imp 123	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ 𝑔:𝑧⟶𝑆) → (𝐺‘𝑔) ∈ 𝑆)
62	24, 61	syldan 280	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → (𝐺‘𝑔) ∈ 𝑆)
63		opexg 4206	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ 𝑆) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
64	49, 62, 63	sylancr 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
65		snidg 3605	. . . . . . . . . 10 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ V → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ {⟨𝑧, (𝐺‘𝑔)⟩})
66		elun2 3290	. . . . . . . . . 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 4809	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ 𝑆) → (⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
69	49, 62, 68	sylancr 411	. . . . . . . . 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 4804	. . . . . . . . . 10 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → dom ℎ = dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
72	71	eleq2d 2236	. . . . . . . . 9 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑧 ∈ dom ℎ ↔ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
73	72	rspcev 2830	. . . . . . . 8 ⊢ (((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴 ∧ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
74	48, 70, 73	syl2anc 409	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
75	15, 74	exlimddv 1886	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
76		eliun 3870	. . . . . 6 ⊢ (𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ ↔ ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
77	75, 76	sylibr 133	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → 𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ)
78	77	ex 114	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝑌 → 𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ))
79	78	ssrdv 3148	. . 3 ⊢ (𝜑 → 𝑌 ⊆ ∪ ℎ ∈ 𝐴 dom ℎ)
80		dmuni 4814	. . . 4 ⊢ dom ∪ 𝐴 = ∪ ℎ ∈ 𝐴 dom ℎ
81	12, 1	tfrcllemssrecs 6320	. . . . 5 ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺))
82		dmss 4803	. . . . 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 3189	. . 3 ⊢ (𝜑 → ∪ ℎ ∈ 𝐴 dom ℎ ⊆ dom recs(𝐺))
85	79, 84	sstrd 3152	. 2 ⊢ (𝜑 → 𝑌 ⊆ dom recs(𝐺))
86	9	dmeqi 4805	. 2 ⊢ dom 𝐹 = dom recs(𝐺)
87	85, 86	sseqtrrdi 3191	1 ⊢ (𝜑 → 𝑌 ⊆ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 968  ∀wal 1341   = wceq 1343  ∃wex 1480   ∈ wcel 2136  {cab 2151  ∀wral 2444  ∃wrex 2445  Vcvv 2726   ∪ cun 3114   ⊆ wss 3116  {csn 3576  ⟨cop 3579  ∪ cuni 3789  ∪ ciun 3866  Ord word 4340  suc csuc 4343  dom cdm 4604   ↾ cres 4606  Fun wfun 5182  ⟶wf 5184  ‘cfv 5188  recscrecs 6272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-recs 6273

This theorem is referenced by:  tfrcldm  6331