Theorem tfrcllembxssdm 6565

Description: Lemma for tfrcl 6573. The union of 𝐵 is defined on all elements of 𝑋. (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)
tfrcllemsucfn.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
tfrcllembacc.3	⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}
tfrcllembacc.u	⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
tfrcllembacc.4	⊢ (𝜑 → 𝐷 ∈ 𝑋)
tfrcllembacc.5	⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))

Assertion

Ref	Expression
tfrcllembxssdm	⊢ (𝜑 → 𝐷 ⊆ dom ∪ 𝐵)

Distinct variable groups:   𝐴,𝑓,𝑔,ℎ,𝑥,𝑦,𝑧   𝐷,𝑓,𝑔,𝑥,𝑦   𝑓,𝐺,𝑥,𝑦   𝑆,𝑓,𝑥,𝑦   𝑓,𝑋,𝑥   𝜑,𝑓,𝑔,ℎ,𝑥,𝑦,𝑧   𝐵,𝑔,ℎ,𝑧   𝑤,𝐵,𝑔,𝑧   𝐷,ℎ,𝑧   ℎ,𝐺,𝑧   𝑤,𝐺,𝑦   𝑆,ℎ,𝑧   𝑧,𝑋

Allowed substitution hints:   𝜑(𝑤)   𝐴(𝑤)   𝐵(𝑥,𝑦,𝑓)   𝐷(𝑤)   𝑆(𝑤,𝑔)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔,ℎ)   𝐺(𝑔)   𝑋(𝑦,𝑤,𝑔,ℎ)

Proof of Theorem tfrcllembxssdm

Step	Hyp	Ref	Expression
1		tfrcllembacc.5	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
2		fveq2 5648	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝑔‘𝑤) = (𝑔‘𝑦))
3		reseq2 5014	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑔 ↾ 𝑤) = (𝑔 ↾ 𝑦))
4	3	fveq2d 5652	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → (𝐺‘(𝑔 ↾ 𝑤)) = (𝐺‘(𝑔 ↾ 𝑦)))
5	2, 4	eqeq12d 2246	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → ((𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)) ↔ (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
6	5	cbvralv 2768	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)) ↔ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))
7	6	anbi2i 457	. . . . . 6 ⊢ ((𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))) ↔ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
8	7	exbii 1654	. . . . 5 ⊢ (∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))) ↔ ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
9	8	ralbii 2539	. . . 4 ⊢ (∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))) ↔ ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
10	1, 9	sylib 122	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
11		simp1 1024	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → 𝜑)
12		simp2 1025	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → 𝑧 ∈ 𝐷)
13		tfrcllembacc.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
14	11, 13	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → 𝐷 ∈ 𝑋)
15		tfrcl.x	. . . . . . . . . . 11 ⊢ (𝜑 → Ord 𝑋)
16		ordtr1 4491	. . . . . . . . . . 11 ⊢ (Ord 𝑋 → ((𝑧 ∈ 𝐷 ∧ 𝐷 ∈ 𝑋) → 𝑧 ∈ 𝑋))
17	15, 16	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑧 ∈ 𝐷 ∧ 𝐷 ∈ 𝑋) → 𝑧 ∈ 𝑋))
18	17	imp 124	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝐷 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
19	11, 12, 14, 18	syl12anc 1272	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → 𝑧 ∈ 𝑋)
20		simp3l 1052	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → 𝑔:𝑧⟶𝑆)
21		feq2 5473	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑓:𝑥⟶𝑆 ↔ 𝑓:𝑧⟶𝑆))
22	21	imbi1d 231	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)))
23	22	albidv 1872	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ ∀𝑓(𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)))
24		tfrcl.ex	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
25	24	3expia 1232	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
26	25	alrimiv 1922	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
27	26	ralrimiva 2606	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
28	27	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
29		simpr 110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
30	23, 28, 29	rspcdva 2916	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑓(𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
31		feq1 5472	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓:𝑧⟶𝑆 ↔ 𝑔:𝑧⟶𝑆))
32		fveq2 5648	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔))
33	32	eleq1d 2300	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ 𝑆 ↔ (𝐺‘𝑔) ∈ 𝑆))
34	31, 33	imbi12d 234	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (𝑔:𝑧⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆)))
35	34	spv 1908	. . . . . . . . . 10 ⊢ (∀𝑓(𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) → (𝑔:𝑧⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆))
36	30, 35	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑔:𝑧⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆))
37	36	imp 124	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑔:𝑧⟶𝑆) → (𝐺‘𝑔) ∈ 𝑆)
38	11, 19, 20, 37	syl21anc 1273	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → (𝐺‘𝑔) ∈ 𝑆)
39		vex 2806	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
40		opexg 4326	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ 𝑆) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
41	39, 38, 40	sylancr 414	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
42		snidg 3702	. . . . . . . . 9 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ V → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ {⟨𝑧, (𝐺‘𝑔)⟩})
43		elun2 3377	. . . . . . . . 9 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ {⟨𝑧, (𝐺‘𝑔)⟩} → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
44	41, 42, 43	3syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
45		simp3r 1053	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))
46		rspe 2582	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → ∃𝑧 ∈ 𝑋 (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
47	19, 20, 45, 46	syl12anc 1272	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → ∃𝑧 ∈ 𝑋 (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
48		feq2 5473	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑔:𝑧⟶𝑆 ↔ 𝑔:𝑥⟶𝑆))
49		raleq 2731	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
50	48, 49	anbi12d 473	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))) ↔ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))))
51	50	cbvrexv 2769	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ 𝑋 (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))) ↔ ∃𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
52	47, 51	sylib 122	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → ∃𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
53		vex 2806	. . . . . . . . . . . 12 ⊢ 𝑔 ∈ V
54		feq1 5472	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑓:𝑥⟶𝑆 ↔ 𝑔:𝑥⟶𝑆))
55		fveq1 5647	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑦) = (𝑔‘𝑦))
56		reseq1 5013	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑔 → (𝑓 ↾ 𝑦) = (𝑔 ↾ 𝑦))
57	56	fveq2d 5652	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑔 → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘(𝑔 ↾ 𝑦)))
58	55, 57	eqeq12d 2246	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
59	58	ralbidv 2533	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
60	54, 59	anbi12d 473	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ((𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))))
61	60	rexbidv 2534	. . . . . . . . . . . 12 ⊢ (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)))))
62		tfrcllemsucfn.1	. . . . . . . . . . . 12 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
63	53, 61, 62	elab2 2955	. . . . . . . . . . 11 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝑋 (𝑔:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))))
64	52, 63	sylibr 134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → 𝑔 ∈ 𝐴)
65	12, 20, 64	3jca 1204	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → (𝑧 ∈ 𝐷 ∧ 𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴))
66		snexg 4280	. . . . . . . . . . 11 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ V → {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V)
67		unexg 4546	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
68	53, 67	mpan 424	. . . . . . . . . . 11 ⊢ ({⟨𝑧, (𝐺‘𝑔)⟩} ∈ V → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
69	41, 66, 68	3syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
70		isset 2810	. . . . . . . . . 10 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V ↔ ∃ℎ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
71	69, 70	sylib 122	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → ∃ℎ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
72		simpr3 1032	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
73		19.8a 1639	. . . . . . . . . . . . . 14 ⊢ ((𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
74		rspe 2582	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐷 ∧ ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
75		tfrcllembacc.3	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}
76	75	abeq2i 2342	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ 𝐵 ↔ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
77	74, 76	sylibr 134	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐷 ∧ ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ℎ ∈ 𝐵)
78	73, 77	sylan2 286	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ℎ ∈ 𝐵)
79	72, 78	eqeltrrd 2309	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐵)
80	79	3exp2 1252	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐷 → (𝑔:𝑧⟶𝑆 → (𝑔 ∈ 𝐴 → (ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐵))))
81	80	3imp 1220	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐷 ∧ 𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴) → (ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐵))
82	81	exlimdv 1867	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐷 ∧ 𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴) → (∃ℎ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐵))
83	65, 71, 82	sylc 62	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐵)
84		elunii 3903	. . . . . . . 8 ⊢ ((⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∧ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐵) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ ∪ 𝐵)
85	44, 83, 84	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ ∪ 𝐵)
86		opeq2 3868	. . . . . . . . . 10 ⊢ (𝑤 = (𝐺‘𝑔) → ⟨𝑧, 𝑤⟩ = ⟨𝑧, (𝐺‘𝑔)⟩)
87	86	eleq1d 2300	. . . . . . . . 9 ⊢ (𝑤 = (𝐺‘𝑔) → (⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵 ↔ ⟨𝑧, (𝐺‘𝑔)⟩ ∈ ∪ 𝐵))
88	87	spcegv 2895	. . . . . . . 8 ⊢ ((𝐺‘𝑔) ∈ 𝑆 → (⟨𝑧, (𝐺‘𝑔)⟩ ∈ ∪ 𝐵 → ∃𝑤⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵))
89	39	eldm2 4935	. . . . . . . 8 ⊢ (𝑧 ∈ dom ∪ 𝐵 ↔ ∃𝑤⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵)
90	88, 89	imbitrrdi 162	. . . . . . 7 ⊢ ((𝐺‘𝑔) ∈ 𝑆 → (⟨𝑧, (𝐺‘𝑔)⟩ ∈ ∪ 𝐵 → 𝑧 ∈ dom ∪ 𝐵))
91	38, 85, 90	sylc 62	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦)))) → 𝑧 ∈ dom ∪ 𝐵)
92	91	3expia 1232	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ((𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))) → 𝑧 ∈ dom ∪ 𝐵))
93	92	exlimdv 1867	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))) → 𝑧 ∈ dom ∪ 𝐵))
94	93	ralimdva 2600	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑦 ∈ 𝑧 (𝑔‘𝑦) = (𝐺‘(𝑔 ↾ 𝑦))) → ∀𝑧 ∈ 𝐷 𝑧 ∈ dom ∪ 𝐵))
95	10, 94	mpd 13	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 𝑧 ∈ dom ∪ 𝐵)
96		dfss3 3217	. 2 ⊢ (𝐷 ⊆ dom ∪ 𝐵 ↔ ∀𝑧 ∈ 𝐷 𝑧 ∈ dom ∪ 𝐵)
97	95, 96	sylibr 134	1 ⊢ (𝜑 → 𝐷 ⊆ dom ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2202  {cab 2217  ∀wral 2511  ∃wrex 2512  Vcvv 2803   ∪ cun 3199   ⊆ wss 3201  {csn 3673  ⟨cop 3676  ∪ cuni 3898  Ord word 4465  suc csuc 4468  dom cdm 4731   ↾ cres 4733  Fun wfun 5327  ⟶wf 5329  ‘cfv 5333  recscrecs 6513

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-tr 4193  df-iord 4469  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341

This theorem is referenced by:  tfrcllembfn  6566