Theorem tfr1onlembfn 6509

Description: Lemma for tfr1on 6515. The union of 𝐵 is a function defined on 𝑥. (Contributed by Jim Kingdon, 15-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
tfr1onlembfn	⊢ (𝜑 → ∪ 𝐵 Fn 𝐷)

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   𝐷,ℎ,𝑧   ℎ,𝐺,𝑧,𝑤,𝑓,𝑦,𝑥   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 tfr1onlembfn

Step	Hyp	Ref	Expression
1		tfr1on.f	. . . . . 6 ⊢ 𝐹 = recs(𝐺)
2		tfr1on.g	. . . . . 6 ⊢ (𝜑 → Fun 𝐺)
3		tfr1on.x	. . . . . 6 ⊢ (𝜑 → Ord 𝑋)
4		tfr1on.ex	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
5		tfr1onlemsucfn.1	. . . . . 6 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
6		tfr1onlembacc.3	. . . . . 6 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}
7		tfr1onlembacc.u	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
8		tfr1onlembacc.4	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
9		tfr1onlembacc.5	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	tfr1onlembacc 6507	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
11	10	unissd 3917	. . . 4 ⊢ (𝜑 → ∪ 𝐵 ⊆ ∪ 𝐴)
12	5, 3	tfr1onlemssrecs 6504	. . . 4 ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺))
13	11, 12	sstrd 3237	. . 3 ⊢ (𝜑 → ∪ 𝐵 ⊆ recs(𝐺))
14		tfrfun 6485	. . 3 ⊢ Fun recs(𝐺)
15		funss 5345	. . 3 ⊢ (∪ 𝐵 ⊆ recs(𝐺) → (Fun recs(𝐺) → Fun ∪ 𝐵))
16	13, 14, 15	mpisyl 1491	. 2 ⊢ (𝜑 → Fun ∪ 𝐵)
17		simpr3 1031	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
18		simpl 109	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝜑)
19	3	adantr 276	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → Ord 𝑋)
20		simpr 110	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝑧 ∈ 𝐷)
21	8	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝐷 ∈ 𝑋)
22	20, 21	jca 306	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝑧 ∈ 𝐷 ∧ 𝐷 ∈ 𝑋))
23		ordtr1 4485	. . . . . . . . . . . . . . . . . . 19 ⊢ (Ord 𝑋 → ((𝑧 ∈ 𝐷 ∧ 𝐷 ∈ 𝑋) → 𝑧 ∈ 𝑋))
24	19, 22, 23	sylc 62	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝑧 ∈ 𝑋)
25	18, 24	jca 306	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝜑 ∧ 𝑧 ∈ 𝑋))
26	2	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → Fun 𝐺)
27	3	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → Ord 𝑋)
28	4	3adant1r 1257	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
29	28	3adant1r 1257	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
30		simplr 529	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑧 ∈ 𝑋)
31		simpr1 1029	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑔 Fn 𝑧)
32		simpr2 1030	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑔 ∈ 𝐴)
33	1, 26, 27, 29, 5, 30, 31, 32	tfr1onlemsucfn 6505	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) Fn suc 𝑧)
34	25, 33	sylan 283	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) Fn suc 𝑧)
35		dffn2 5484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) Fn suc 𝑧 ↔ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶V)
36	34, 35	sylib 122	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶V)
37		fssxp 5502	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶V → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ⊆ (suc 𝑧 × V))
38	36, 37	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ⊆ (suc 𝑧 × V))
39		ordelon 4480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Ord 𝑋 ∧ 𝐷 ∈ 𝑋) → 𝐷 ∈ On)
40	3, 8, 39	syl2anc 411	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷 ∈ On)
41		eloni 4472	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ On → Ord 𝐷)
42	40, 41	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Ord 𝐷)
43	42	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → Ord 𝐷)
44		simplr 529	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑧 ∈ 𝐷)
45		ordsucss 4602	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝐷 → (𝑧 ∈ 𝐷 → suc 𝑧 ⊆ 𝐷))
46	43, 44, 45	sylc 62	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → suc 𝑧 ⊆ 𝐷)
47		xpss1 4836	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑧 ⊆ 𝐷 → (suc 𝑧 × V) ⊆ (𝐷 × V))
48	46, 47	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (suc 𝑧 × V) ⊆ (𝐷 × V))
49	38, 48	sstrd 3237	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ⊆ (𝐷 × V))
50		vex 2805	. . . . . . . . . . . . . . 15 ⊢ 𝑔 ∈ V
51		vex 2805	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
52	18	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝜑)
53	24	adantr 276	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑧 ∈ 𝑋)
54		simpr1 1029	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑔 Fn 𝑧)
55		fneq2 5419	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → (𝑓 Fn 𝑥 ↔ 𝑓 Fn 𝑧))
56	55	imbi1d 231	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V)))
57	56	albidv 1872	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → (∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V) ↔ ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V)))
58	4	3expia 1231	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
59	58	alrimiv 1922	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
60	59	ralrimiva 2605	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
61	60	3ad2ant1 1044	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑔 Fn 𝑧) → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
62		simp2 1024	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑔 Fn 𝑧) → 𝑧 ∈ 𝑋)
63	57, 61, 62	rspcdva 2915	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑔 Fn 𝑧) → ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V))
64		simp3 1025	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑔 Fn 𝑧) → 𝑔 Fn 𝑧)
65		fneq1 5418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑧 ↔ 𝑔 Fn 𝑧))
66		fveq2 5639	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔))
67	66	eleq1d 2300	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ V ↔ (𝐺‘𝑔) ∈ V))
68	65, 67	imbi12d 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺‘𝑔) ∈ V)))
69	68	spv 1908	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺‘𝑔) ∈ V))
70	63, 64, 69	sylc 62	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑔 Fn 𝑧) → (𝐺‘𝑔) ∈ V)
71	52, 53, 54, 70	syl3anc 1273	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝐺‘𝑔) ∈ V)
72		opexg 4320	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ V) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
73	51, 71, 72	sylancr 414	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
74		snexg 4274	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ V → {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V)
75	73, 74	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V)
76		unexg 4540	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
77	50, 75, 76	sylancr 414	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
78		elpwg 3660	. . . . . . . . . . . . . 14 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝒫 (𝐷 × V) ↔ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ⊆ (𝐷 × V)))
79	77, 78	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝒫 (𝐷 × V) ↔ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ⊆ (𝐷 × V)))
80	49, 79	mpbird 167	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝒫 (𝐷 × V))
81	17, 80	eqeltrd 2308	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ℎ ∈ 𝒫 (𝐷 × V))
82	81	ex 115	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ((𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝐷 × V)))
83	82	exlimdv 1867	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝐷 × V)))
84	83	rexlimdva 2650	. . . . . . . 8 ⊢ (𝜑 → (∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝐷 × V)))
85	84	abssdv 3301	. . . . . . 7 ⊢ (𝜑 → {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))} ⊆ 𝒫 (𝐷 × V))
86	6, 85	eqsstrid 3273	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝒫 (𝐷 × V))
87		sspwuni 4055	. . . . . 6 ⊢ (𝐵 ⊆ 𝒫 (𝐷 × V) ↔ ∪ 𝐵 ⊆ (𝐷 × V))
88	86, 87	sylib 122	. . . . 5 ⊢ (𝜑 → ∪ 𝐵 ⊆ (𝐷 × V))
89		dmss 4930	. . . . 5 ⊢ (∪ 𝐵 ⊆ (𝐷 × V) → dom ∪ 𝐵 ⊆ dom (𝐷 × V))
90	88, 89	syl 14	. . . 4 ⊢ (𝜑 → dom ∪ 𝐵 ⊆ dom (𝐷 × V))
91		dmxpss 5167	. . . 4 ⊢ dom (𝐷 × V) ⊆ 𝐷
92	90, 91	sstrdi 3239	. . 3 ⊢ (𝜑 → dom ∪ 𝐵 ⊆ 𝐷)
93	1, 2, 3, 4, 5, 6, 7, 8, 9	tfr1onlembxssdm 6508	. . 3 ⊢ (𝜑 → 𝐷 ⊆ dom ∪ 𝐵)
94	92, 93	eqssd 3244	. 2 ⊢ (𝜑 → dom ∪ 𝐵 = 𝐷)
95		df-fn 5329	. 2 ⊢ (∪ 𝐵 Fn 𝐷 ↔ (Fun ∪ 𝐵 ∧ dom ∪ 𝐵 = 𝐷))
96	16, 94, 95	sylanbrc 417	1 ⊢ (𝜑 → ∪ 𝐵 Fn 𝐷)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004  ∀wal 1395   = wceq 1397  ∃wex 1540   ∈ wcel 2202  {cab 2217  ∀wral 2510  ∃wrex 2511  Vcvv 2802   ∪ cun 3198   ⊆ wss 3200  𝒫 cpw 3652  {csn 3669  ⟨cop 3672  ∪ cuni 3893  Ord word 4459  Oncon0 4460  suc csuc 4462   × cxp 4723  dom cdm 4725   ↾ cres 4727  Fun wfun 5320   Fn wfn 5321  ⟶wf 5322  ‘cfv 5326  recscrecs 6469

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-recs 6470

This theorem is referenced by:  tfr1onlembex  6510  tfr1onlemubacc  6511  tfr1onlemex  6512