Theorem tfr1onlembfn 6303

Description: Lemma for tfr1on 6309. 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 6301	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
11	10	unissd 3807	. . . 4 ⊢ (𝜑 → ∪ 𝐵 ⊆ ∪ 𝐴)
12	5, 3	tfr1onlemssrecs 6298	. . . 4 ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺))
13	11, 12	sstrd 3147	. . 3 ⊢ (𝜑 → ∪ 𝐵 ⊆ recs(𝐺))
14		tfrfun 6279	. . 3 ⊢ Fun recs(𝐺)
15		funss 5201	. . 3 ⊢ (∪ 𝐵 ⊆ recs(𝐺) → (Fun recs(𝐺) → Fun ∪ 𝐵))
16	13, 14, 15	mpisyl 1433	. 2 ⊢ (𝜑 → Fun ∪ 𝐵)
17		simpr3 994	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
18		simpl 108	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝜑)
19	3	adantr 274	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → Ord 𝑋)
20		simpr 109	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝑧 ∈ 𝐷)
21	8	adantr 274	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → 𝐷 ∈ 𝑋)
22	20, 21	jca 304	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝑧 ∈ 𝐷 ∧ 𝐷 ∈ 𝑋))
23		ordtr1 4360	. . . . . . . . . . . . . . . . . . 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 304	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝜑 ∧ 𝑧 ∈ 𝑋))
26	2	ad2antrr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → Fun 𝐺)
27	3	ad2antrr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → Ord 𝑋)
28	4	3adant1r 1220	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
29	28	3adant1r 1220	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
30		simplr 520	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑧 ∈ 𝑋)
31		simpr1 992	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑔 Fn 𝑧)
32		simpr2 993	. . . . . . . . . . . . . . . . . 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 6299	. . . . . . . . . . . . . . . . 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 281	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) Fn suc 𝑧)
35		dffn2 5333	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) Fn suc 𝑧 ↔ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶V)
36	34, 35	sylib 121	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶V)
37		fssxp 5349	. . . . . . . . . . . . . . 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 4355	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Ord 𝑋 ∧ 𝐷 ∈ 𝑋) → 𝐷 ∈ On)
40	3, 8, 39	syl2anc 409	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐷 ∈ On)
41		eloni 4347	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ On → Ord 𝐷)
42	40, 41	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → Ord 𝐷)
43	42	ad2antrr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → Ord 𝐷)
44		simplr 520	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑧 ∈ 𝐷)
45		ordsucss 4475	. . . . . . . . . . . . . . . 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 4708	. . . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ⊆ (𝐷 × V))
50		vex 2724	. . . . . . . . . . . . . . 15 ⊢ 𝑔 ∈ V
51		vex 2724	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
52	18	adantr 274	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝜑)
53	24	adantr 274	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑧 ∈ 𝑋)
54		simpr1 992	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑔 Fn 𝑧)
55		fneq2 5271	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → (𝑓 Fn 𝑥 ↔ 𝑓 Fn 𝑧))
56	55	imbi1d 230	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V)))
57	56	albidv 1811	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → (∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V) ↔ ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V)))
58	4	3expia 1194	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
59	58	alrimiv 1861	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
60	59	ralrimiva 2537	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
61	60	3ad2ant1 1007	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑔 Fn 𝑧) → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
62		simp2 987	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑔 Fn 𝑧) → 𝑧 ∈ 𝑋)
63	57, 61, 62	rspcdva 2830	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑔 Fn 𝑧) → ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V))
64		simp3 988	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑔 Fn 𝑧) → 𝑔 Fn 𝑧)
65		fneq1 5270	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑧 ↔ 𝑔 Fn 𝑧))
66		fveq2 5480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔))
67	66	eleq1d 2233	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ V ↔ (𝐺‘𝑔) ∈ V))
68	65, 67	imbi12d 233	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺‘𝑔) ∈ V)))
69	68	spv 1847	. . . . . . . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝐺‘𝑔) ∈ V)
72		opexg 4200	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ V) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
73	51, 71, 72	sylancr 411	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
74		snexg 4157	. . . . . . . . . . . . . . . 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 4415	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
77	50, 75, 76	sylancr 411	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
78		elpwg 3561	. . . . . . . . . . . . . 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 166	. . . . . . . . . . . 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 2241	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ℎ ∈ 𝒫 (𝐷 × V))
82	81	ex 114	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ((𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝐷 × V)))
83	82	exlimdv 1806	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝐷 × V)))
84	83	rexlimdva 2581	. . . . . . . 8 ⊢ (𝜑 → (∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝐷 × V)))
85	84	abssdv 3211	. . . . . . 7 ⊢ (𝜑 → {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))} ⊆ 𝒫 (𝐷 × V))
86	6, 85	eqsstrid 3183	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝒫 (𝐷 × V))
87		sspwuni 3944	. . . . . 6 ⊢ (𝐵 ⊆ 𝒫 (𝐷 × V) ↔ ∪ 𝐵 ⊆ (𝐷 × V))
88	86, 87	sylib 121	. . . . 5 ⊢ (𝜑 → ∪ 𝐵 ⊆ (𝐷 × V))
89		dmss 4797	. . . . 5 ⊢ (∪ 𝐵 ⊆ (𝐷 × V) → dom ∪ 𝐵 ⊆ dom (𝐷 × V))
90	88, 89	syl 14	. . . 4 ⊢ (𝜑 → dom ∪ 𝐵 ⊆ dom (𝐷 × V))
91		dmxpss 5028	. . . 4 ⊢ dom (𝐷 × V) ⊆ 𝐷
92	90, 91	sstrdi 3149	. . 3 ⊢ (𝜑 → dom ∪ 𝐵 ⊆ 𝐷)
93	1, 2, 3, 4, 5, 6, 7, 8, 9	tfr1onlembxssdm 6302	. . 3 ⊢ (𝜑 → 𝐷 ⊆ dom ∪ 𝐵)
94	92, 93	eqssd 3154	. 2 ⊢ (𝜑 → dom ∪ 𝐵 = 𝐷)
95		df-fn 5185	. 2 ⊢ (∪ 𝐵 Fn 𝐷 ↔ (Fun ∪ 𝐵 ∧ dom ∪ 𝐵 = 𝐷))
96	16, 94, 95	sylanbrc 414	1 ⊢ (𝜑 → ∪ 𝐵 Fn 𝐷)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 967  ∀wal 1340   = wceq 1342  ∃wex 1479   ∈ wcel 2135  {cab 2150  ∀wral 2442  ∃wrex 2443  Vcvv 2721   ∪ cun 3109   ⊆ wss 3111  𝒫 cpw 3553  {csn 3570  ⟨cop 3573  ∪ cuni 3783  Ord word 4334  Oncon0 4335  suc csuc 4337   × cxp 4596  dom cdm 4598   ↾ cres 4600  Fun wfun 5176   Fn wfn 5177  ⟶wf 5178  ‘cfv 5182  recscrecs 6263

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-iord 4338  df-on 4340  df-suc 4343  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-fv 5190  df-recs 6264

This theorem is referenced by:  tfr1onlembex  6304  tfr1onlemubacc  6305  tfr1onlemex  6306