Theorem tfr1onlembxssdm 6396

Description: Lemma for tfr1on 6403. The union of 𝐵 is defined on all elements of 𝑋. (Contributed by Jim Kingdon, 14-Mar-2022.)

Hypotheses

Ref	Expression
tfr1on.f	⊢ 𝐹 = recs(𝐺)
tfr1on.g	⊢ (𝜑 → Fun 𝐺)
tfr1on.x	⊢ (𝜑 → Ord 𝑋)
tfr1on.ex	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
tfr1onlemsucfn.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
tfr1onlembacc.3	⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}
tfr1onlembacc.u	⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
tfr1onlembacc.4	⊢ (𝜑 → 𝐷 ∈ 𝑋)
tfr1onlembacc.5	⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))

Assertion

Ref	Expression
tfr1onlembxssdm	⊢ (𝜑 → 𝐷 ⊆ dom ∪ 𝐵)

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

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

Proof of Theorem tfr1onlembxssdm

Step	Hyp	Ref	Expression
1		tfr1onlembacc.5	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
2		simp1 999	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → 𝜑)
3		simp2 1000	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → 𝑧 ∈ 𝐷)
4		tfr1onlembacc.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
5	2, 4	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → 𝐷 ∈ 𝑋)
6		tfr1on.x	. . . . . . . . . . 11 ⊢ (𝜑 → Ord 𝑋)
7		ordtr1 4419	. . . . . . . . . . 11 ⊢ (Ord 𝑋 → ((𝑧 ∈ 𝐷 ∧ 𝐷 ∈ 𝑋) → 𝑧 ∈ 𝑋))
8	6, 7	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑧 ∈ 𝐷 ∧ 𝐷 ∈ 𝑋) → 𝑧 ∈ 𝑋))
9	8	imp 124	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐷 ∧ 𝐷 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
10	2, 3, 5, 9	syl12anc 1247	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → 𝑧 ∈ 𝑋)
11		simp3l 1027	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → 𝑔 Fn 𝑧)
12		fneq2 5343	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑓 Fn 𝑥 ↔ 𝑓 Fn 𝑧))
13	12	imbi1d 231	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V)))
14	13	albidv 1835	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V) ↔ ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V)))
15		tfr1on.ex	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
16	15	3expia 1207	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
17	16	alrimiv 1885	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
18	17	ralrimiva 2567	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
19	18	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
20		simpr 110	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
21	14, 19, 20	rspcdva 2869	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V))
22		fneq1 5342	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑧 ↔ 𝑔 Fn 𝑧))
23		fveq2 5554	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔))
24	23	eleq1d 2262	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ V ↔ (𝐺‘𝑔) ∈ V))
25	22, 24	imbi12d 234	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺‘𝑔) ∈ V)))
26	25	spv 1871	. . . . . . . . . 10 ⊢ (∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺‘𝑔) ∈ V))
27	21, 26	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → (𝑔 Fn 𝑧 → (𝐺‘𝑔) ∈ V))
28	27	imp 124	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑔 Fn 𝑧) → (𝐺‘𝑔) ∈ V)
29	2, 10, 11, 28	syl21anc 1248	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → (𝐺‘𝑔) ∈ V)
30		vex 2763	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
31		opexg 4257	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ V) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
32	30, 29, 31	sylancr 414	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
33		snidg 3647	. . . . . . . . 9 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ V → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ {⟨𝑧, (𝐺‘𝑔)⟩})
34		elun2 3327	. . . . . . . . 9 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ {⟨𝑧, (𝐺‘𝑔)⟩} → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
35	32, 33, 34	3syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
36		simp3r 1028	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))
37		rspe 2543	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → ∃𝑧 ∈ 𝑋 (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
38	10, 11, 36, 37	syl12anc 1247	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → ∃𝑧 ∈ 𝑋 (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
39		vex 2763	. . . . . . . . . . . 12 ⊢ 𝑔 ∈ V
40		tfr1onlemsucfn.1	. . . . . . . . . . . . 13 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
41	40	tfr1onlem3ag 6390	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ V → (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ 𝑋 (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))))
42	39, 41	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ 𝑋 (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
43	38, 42	sylibr 134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → 𝑔 ∈ 𝐴)
44	3, 11, 43	3jca 1179	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → (𝑧 ∈ 𝐷 ∧ 𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴))
45		snexg 4213	. . . . . . . . . . 11 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ V → {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V)
46		unexg 4474	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
47	39, 46	mpan 424	. . . . . . . . . . 11 ⊢ ({⟨𝑧, (𝐺‘𝑔)⟩} ∈ V → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
48	32, 45, 47	3syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
49		isset 2766	. . . . . . . . . 10 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V ↔ ∃ℎ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
50	48, 49	sylib 122	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → ∃ℎ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
51		simpr3 1007	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
52		19.8a 1601	. . . . . . . . . . . . . 14 ⊢ ((𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
53		rspe 2543	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐷 ∧ ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
54		tfr1onlembacc.3	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}
55	54	abeq2i 2304	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ 𝐵 ↔ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
56	53, 55	sylibr 134	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐷 ∧ ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ℎ ∈ 𝐵)
57	52, 56	sylan2 286	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ℎ ∈ 𝐵)
58	51, 57	eqeltrrd 2271	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐵)
59	58	3exp2 1227	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐷 → (𝑔 Fn 𝑧 → (𝑔 ∈ 𝐴 → (ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐵))))
60	59	3imp 1195	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐷 ∧ 𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴) → (ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐵))
61	60	exlimdv 1830	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐷 ∧ 𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴) → (∃ℎ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐵))
62	44, 50, 61	sylc 62	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐵)
63		elunii 3840	. . . . . . . 8 ⊢ ((⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∧ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐵) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ ∪ 𝐵)
64	35, 62, 63	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ ∪ 𝐵)
65		opeq2 3805	. . . . . . . . . 10 ⊢ (𝑤 = (𝐺‘𝑔) → ⟨𝑧, 𝑤⟩ = ⟨𝑧, (𝐺‘𝑔)⟩)
66	65	eleq1d 2262	. . . . . . . . 9 ⊢ (𝑤 = (𝐺‘𝑔) → (⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵 ↔ ⟨𝑧, (𝐺‘𝑔)⟩ ∈ ∪ 𝐵))
67	66	spcegv 2848	. . . . . . . 8 ⊢ ((𝐺‘𝑔) ∈ V → (⟨𝑧, (𝐺‘𝑔)⟩ ∈ ∪ 𝐵 → ∃𝑤⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵))
68	30	eldm2 4860	. . . . . . . 8 ⊢ (𝑧 ∈ dom ∪ 𝐵 ↔ ∃𝑤⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵)
69	67, 68	imbitrrdi 162	. . . . . . 7 ⊢ ((𝐺‘𝑔) ∈ V → (⟨𝑧, (𝐺‘𝑔)⟩ ∈ ∪ 𝐵 → 𝑧 ∈ dom ∪ 𝐵))
70	29, 64, 69	sylc 62	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) → 𝑧 ∈ dom ∪ 𝐵)
71	70	3expia 1207	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))) → 𝑧 ∈ dom ∪ 𝐵))
72	71	exlimdv 1830	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))) → 𝑧 ∈ dom ∪ 𝐵))
73	72	ralimdva 2561	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))) → ∀𝑧 ∈ 𝐷 𝑧 ∈ dom ∪ 𝐵))
74	1, 73	mpd 13	. 2 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 𝑧 ∈ dom ∪ 𝐵)
75		dfss3 3169	. 2 ⊢ (𝐷 ⊆ dom ∪ 𝐵 ↔ ∀𝑧 ∈ 𝐷 𝑧 ∈ dom ∪ 𝐵)
76	74, 75	sylibr 134	1 ⊢ (𝜑 → 𝐷 ⊆ dom ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980  ∀wal 1362   = wceq 1364  ∃wex 1503   ∈ wcel 2164  {cab 2179  ∀wral 2472  ∃wrex 2473  Vcvv 2760   ∪ cun 3151   ⊆ wss 3153  {csn 3618  ⟨cop 3621  ∪ cuni 3835  Ord word 4393  suc csuc 4396  dom cdm 4659   ↾ cres 4661  Fun wfun 5248   Fn wfn 5249  ‘cfv 5254  recscrecs 6357

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-tr 4128  df-iord 4397  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by:  tfr1onlembfn  6397