Theorem tfrcllembfn 6336

Description: Lemma for tfrcl 6343. The union of 𝐵 is a function defined on 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
tfrcllembfn	⊢ (𝜑 → ∪ 𝐵:𝐷⟶𝑆)

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,𝑔,ℎ,𝑧   𝑧,𝑋

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

Proof of Theorem tfrcllembfn

Step	Hyp	Ref	Expression
1		tfrcl.f	. . . . . . 7 ⊢ 𝐹 = recs(𝐺)
2		tfrcl.g	. . . . . . 7 ⊢ (𝜑 → Fun 𝐺)
3		tfrcl.x	. . . . . . 7 ⊢ (𝜑 → Ord 𝑋)
4		tfrcl.ex	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
5		tfrcllemsucfn.1	. . . . . . 7 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
6		tfrcllembacc.3	. . . . . . 7 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}
7		tfrcllembacc.u	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
8		tfrcllembacc.4	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
9		tfrcllembacc.5	. . . . . . 7 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	tfrcllembacc 6334	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
11	10	unissd 3820	. . . . 5 ⊢ (𝜑 → ∪ 𝐵 ⊆ ∪ 𝐴)
12	5, 3	tfrcllemssrecs 6331	. . . . 5 ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺))
13	11, 12	sstrd 3157	. . . 4 ⊢ (𝜑 → ∪ 𝐵 ⊆ recs(𝐺))
14		tfrfun 6299	. . . 4 ⊢ Fun recs(𝐺)
15		funss 5217	. . . 4 ⊢ (∪ 𝐵 ⊆ recs(𝐺) → (Fun recs(𝐺) → Fun ∪ 𝐵))
16	13, 14, 15	mpisyl 1439	. . 3 ⊢ (𝜑 → Fun ∪ 𝐵)
17		simpr3 1000	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 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 4373	. . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → Fun 𝐺)
27	3	ad2antrr 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → Ord 𝑋)
28	4	3adant1r 1226	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
29	28	3adant1r 1226	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
30		simplr 525	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑧 ∈ 𝑋)
31		simpr1 998	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑔:𝑧⟶𝑆)
32		simpr2 999	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑔 ∈ 𝐴)
33	1, 26, 27, 29, 5, 30, 31, 32	tfrcllemsucfn 6332	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶𝑆)
34	25, 33	sylan 281	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶𝑆)
35		fssxp 5365	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}):suc 𝑧⟶𝑆 → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ⊆ (suc 𝑧 × 𝑆))
36	34, 35	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ⊆ (suc 𝑧 × 𝑆))
37		ordelon 4368	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Ord 𝑋 ∧ 𝐷 ∈ 𝑋) → 𝐷 ∈ On)
38	3, 8, 37	syl2anc 409	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐷 ∈ On)
39		eloni 4360	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 ∈ On → Ord 𝐷)
40	38, 39	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Ord 𝐷)
41	40	ad2antrr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → Ord 𝐷)
42		simplr 525	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑧 ∈ 𝐷)
43		ordsucss 4488	. . . . . . . . . . . . . . . . 17 ⊢ (Ord 𝐷 → (𝑧 ∈ 𝐷 → suc 𝑧 ⊆ 𝐷))
44	41, 42, 43	sylc 62	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → suc 𝑧 ⊆ 𝐷)
45		xpss1 4721	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑧 ⊆ 𝐷 → (suc 𝑧 × 𝑆) ⊆ (𝐷 × 𝑆))
46	44, 45	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (suc 𝑧 × 𝑆) ⊆ (𝐷 × 𝑆))
47	36, 46	sstrd 3157	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ⊆ (𝐷 × 𝑆))
48		vex 2733	. . . . . . . . . . . . . . . 16 ⊢ 𝑔 ∈ V
49		vex 2733	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
50	18	adantr 274	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝜑)
51	24	adantr 274	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑧 ∈ 𝑋)
52		simpr1 998	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑔:𝑧⟶𝑆)
53		feq2 5331	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑧 → (𝑓:𝑥⟶𝑆 ↔ 𝑓:𝑧⟶𝑆))
54	53	imbi1d 230	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑧 → ((𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)))
55	54	albidv 1817	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑧 → (∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ ∀𝑓(𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)))
56	4	3expia 1200	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
57	56	alrimiv 1867	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
58	57	ralrimiva 2543	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
59	58	3ad2ant1 1013	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑔:𝑧⟶𝑆) → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
60		simp2 993	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑔:𝑧⟶𝑆) → 𝑧 ∈ 𝑋)
61	55, 59, 60	rspcdva 2839	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑔:𝑧⟶𝑆) → ∀𝑓(𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))
62		simp3 994	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑔:𝑧⟶𝑆) → 𝑔:𝑧⟶𝑆)
63		feq1 5330	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑔 → (𝑓:𝑧⟶𝑆 ↔ 𝑔:𝑧⟶𝑆))
64		fveq2 5496	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔))
65	64	eleq1d 2239	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ 𝑆 ↔ (𝐺‘𝑔) ∈ 𝑆))
66	63, 65	imbi12d 233	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 = 𝑔 → ((𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (𝑔:𝑧⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆)))
67	66	spv 1853	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑓(𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) → (𝑔:𝑧⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆))
68	61, 62, 67	sylc 62	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋 ∧ 𝑔:𝑧⟶𝑆) → (𝐺‘𝑔) ∈ 𝑆)
69	50, 51, 52, 68	syl3anc 1233	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝐺‘𝑔) ∈ 𝑆)
70		opexg 4213	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ 𝑆) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
71	49, 69, 70	sylancr 412	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
72		snexg 4170	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ V → {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V)
73	71, 72	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V)
74		unexg 4428	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺‘𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
75	48, 73, 74	sylancr 412	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V)
76		elpwg 3574	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ V → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝒫 (𝐷 × 𝑆) ↔ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ⊆ (𝐷 × 𝑆)))
77	75, 76	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝒫 (𝐷 × 𝑆) ↔ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ⊆ (𝐷 × 𝑆)))
78	47, 77	mpbird 166	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝒫 (𝐷 × 𝑆))
79	17, 78	eqeltrd 2247	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ℎ ∈ 𝒫 (𝐷 × 𝑆))
80	79	ex 114	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ((𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝐷 × 𝑆)))
81	80	exlimdv 1812	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝐷 × 𝑆)))
82	81	rexlimdva 2587	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝐷 × 𝑆)))
83	82	abssdv 3221	. . . . . . . 8 ⊢ (𝜑 → {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))} ⊆ 𝒫 (𝐷 × 𝑆))
84	6, 83	eqsstrid 3193	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝒫 (𝐷 × 𝑆))
85		sspwuni 3957	. . . . . . 7 ⊢ (𝐵 ⊆ 𝒫 (𝐷 × 𝑆) ↔ ∪ 𝐵 ⊆ (𝐷 × 𝑆))
86	84, 85	sylib 121	. . . . . 6 ⊢ (𝜑 → ∪ 𝐵 ⊆ (𝐷 × 𝑆))
87		dmss 4810	. . . . . 6 ⊢ (∪ 𝐵 ⊆ (𝐷 × 𝑆) → dom ∪ 𝐵 ⊆ dom (𝐷 × 𝑆))
88	86, 87	syl 14	. . . . 5 ⊢ (𝜑 → dom ∪ 𝐵 ⊆ dom (𝐷 × 𝑆))
89		dmxpss 5041	. . . . 5 ⊢ dom (𝐷 × 𝑆) ⊆ 𝐷
90	88, 89	sstrdi 3159	. . . 4 ⊢ (𝜑 → dom ∪ 𝐵 ⊆ 𝐷)
91	1, 2, 3, 4, 5, 6, 7, 8, 9	tfrcllembxssdm 6335	. . . 4 ⊢ (𝜑 → 𝐷 ⊆ dom ∪ 𝐵)
92	90, 91	eqssd 3164	. . 3 ⊢ (𝜑 → dom ∪ 𝐵 = 𝐷)
93		df-fn 5201	. . 3 ⊢ (∪ 𝐵 Fn 𝐷 ↔ (Fun ∪ 𝐵 ∧ dom ∪ 𝐵 = 𝐷))
94	16, 92, 93	sylanbrc 415	. 2 ⊢ (𝜑 → ∪ 𝐵 Fn 𝐷)
95		rnss 4841	. . . 4 ⊢ (∪ 𝐵 ⊆ (𝐷 × 𝑆) → ran ∪ 𝐵 ⊆ ran (𝐷 × 𝑆))
96	86, 95	syl 14	. . 3 ⊢ (𝜑 → ran ∪ 𝐵 ⊆ ran (𝐷 × 𝑆))
97		rnxpss 5042	. . 3 ⊢ ran (𝐷 × 𝑆) ⊆ 𝑆
98	96, 97	sstrdi 3159	. 2 ⊢ (𝜑 → ran ∪ 𝐵 ⊆ 𝑆)
99		df-f 5202	. 2 ⊢ (∪ 𝐵:𝐷⟶𝑆 ↔ (∪ 𝐵 Fn 𝐷 ∧ ran ∪ 𝐵 ⊆ 𝑆))
100	94, 98, 99	sylanbrc 415	1 ⊢ (𝜑 → ∪ 𝐵:𝐷⟶𝑆)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 973  ∀wal 1346   = wceq 1348  ∃wex 1485   ∈ wcel 2141  {cab 2156  ∀wral 2448  ∃wrex 2449  Vcvv 2730   ∪ cun 3119   ⊆ wss 3121  𝒫 cpw 3566  {csn 3583  ⟨cop 3586  ∪ cuni 3796  Ord word 4347  Oncon0 4348  suc csuc 4350   × cxp 4609  dom cdm 4611  ran crn 4612   ↾ cres 4613  Fun wfun 5192   Fn wfn 5193  ⟶wf 5194  ‘cfv 5198  recscrecs 6283

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-recs 6284

This theorem is referenced by:  tfrcllembex  6337  tfrcllemubacc  6338  tfrcllemex  6339