Theorem ptuncnv 21610

Description: Exhibit the converse function of the map 𝐺 which joins two product topologies on disjoint index sets. (Contributed by Mario Carneiro, 8-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Hypotheses

Ref	Expression
ptunhmeo.x	⊢ 𝑋 = ∪ 𝐾
ptunhmeo.y	⊢ 𝑌 = ∪ 𝐿
ptunhmeo.j	⊢ 𝐽 = (∏t‘𝐹)
ptunhmeo.k	⊢ 𝐾 = (∏t‘(𝐹 ↾ 𝐴))
ptunhmeo.l	⊢ 𝐿 = (∏t‘(𝐹 ↾ 𝐵))
ptunhmeo.g	⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥 ∪ 𝑦))
ptunhmeo.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
ptunhmeo.f	⊢ (𝜑 → 𝐹:𝐶⟶Top)
ptunhmeo.u	⊢ (𝜑 → 𝐶 = (𝐴 ∪ 𝐵))
ptunhmeo.i	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)

Assertion

Ref	Expression
ptuncnv	⊢ (𝜑 → ◡𝐺 = (𝑧 ∈ ∪ 𝐽 ↦ ⟨(𝑧 ↾ 𝐴), (𝑧 ↾ 𝐵)⟩))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑧,𝐺   𝜑,𝑥,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐽,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝑥,𝐿,𝑦,𝑧   𝑧,𝑉   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧

Allowed substitution hints:   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem ptuncnv

Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptunhmeo.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥 ∪ 𝑦))
2		vex 3203	. . . . . . 7 ⊢ 𝑥 ∈ V
3		vex 3203	. . . . . . 7 ⊢ 𝑦 ∈ V
4	2, 3	op1std 7178	. . . . . 6 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑤) = 𝑥)
5	2, 3	op2ndd 7179	. . . . . 6 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑤) = 𝑦)
6	4, 5	uneq12d 3768	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ((1st ‘𝑤) ∪ (2nd ‘𝑤)) = (𝑥 ∪ 𝑦))
7	6	mpt2mpt 6752	. . . 4 ⊢ (𝑤 ∈ (𝑋 × 𝑌) ↦ ((1st ‘𝑤) ∪ (2nd ‘𝑤))) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝑥 ∪ 𝑦))
8	1, 7	eqtr4i 2647	. . 3 ⊢ 𝐺 = (𝑤 ∈ (𝑋 × 𝑌) ↦ ((1st ‘𝑤) ∪ (2nd ‘𝑤)))
9		xp1st 7198	. . . . . . 7 ⊢ (𝑤 ∈ (𝑋 × 𝑌) → (1st ‘𝑤) ∈ 𝑋)
10	9	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑋 × 𝑌)) → (1st ‘𝑤) ∈ 𝑋)
11		ixpeq2 7922	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ 𝐴 ∪ ((𝐹 ↾ 𝐴)‘𝑘) = ∪ (𝐹‘𝑘) → X𝑘 ∈ 𝐴 ∪ ((𝐹 ↾ 𝐴)‘𝑘) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
12		fvres 6207	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑘) = (𝐹‘𝑘))
13	12	unieqd 4446	. . . . . . . . . 10 ⊢ (𝑘 ∈ 𝐴 → ∪ ((𝐹 ↾ 𝐴)‘𝑘) = ∪ (𝐹‘𝑘))
14	11, 13	mprg 2926	. . . . . . . . 9 ⊢ X𝑘 ∈ 𝐴 ∪ ((𝐹 ↾ 𝐴)‘𝑘) = X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘)
15		ptunhmeo.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
16		ssun1 3776	. . . . . . . . . . . 12 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
17		ptunhmeo.u	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 = (𝐴 ∪ 𝐵))
18	16, 17	syl5sseqr 3654	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
19	15, 18	ssexd 4805	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ V)
20		ptunhmeo.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝐶⟶Top)
21	20, 18	fssresd 6071	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ↾ 𝐴):𝐴⟶Top)
22		ptunhmeo.k	. . . . . . . . . . 11 ⊢ 𝐾 = (∏t‘(𝐹 ↾ 𝐴))
23	22	ptuni 21397	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ (𝐹 ↾ 𝐴):𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ ((𝐹 ↾ 𝐴)‘𝑘) = ∪ 𝐾)
24	19, 21, 23	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → X𝑘 ∈ 𝐴 ∪ ((𝐹 ↾ 𝐴)‘𝑘) = ∪ 𝐾)
25	14, 24	syl5eqr 2670	. . . . . . . 8 ⊢ (𝜑 → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐾)
26		ptunhmeo.x	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐾
27	25, 26	syl6eqr 2674	. . . . . . 7 ⊢ (𝜑 → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = 𝑋)
28	27	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑋 × 𝑌)) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = 𝑋)
29	10, 28	eleqtrrd 2704	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑋 × 𝑌)) → (1st ‘𝑤) ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
30		xp2nd 7199	. . . . . . 7 ⊢ (𝑤 ∈ (𝑋 × 𝑌) → (2nd ‘𝑤) ∈ 𝑌)
31	30	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑋 × 𝑌)) → (2nd ‘𝑤) ∈ 𝑌)
32	17	eqcomd 2628	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∪ 𝐵) = 𝐶)
33		ptunhmeo.i	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
34		uneqdifeq 4057	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) = 𝐶 ↔ (𝐶 ∖ 𝐴) = 𝐵))
35	18, 33, 34	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) = 𝐶 ↔ (𝐶 ∖ 𝐴) = 𝐵))
36	32, 35	mpbid 222	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∖ 𝐴) = 𝐵)
37	36	ixpeq1d 7920	. . . . . . . 8 ⊢ (𝜑 → X𝑘 ∈ (𝐶 ∖ 𝐴)∪ (𝐹‘𝑘) = X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘))
38		ixpeq2 7922	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ (𝐹‘𝑘) → X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘))
39		fvres 6207	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑘) = (𝐹‘𝑘))
40	39	unieqd 4446	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐵 → ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ (𝐹‘𝑘))
41	38, 40	mprg 2926	. . . . . . . . . 10 ⊢ X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘)
42		ssun2 3777	. . . . . . . . . . . . 13 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
43	42, 17	syl5sseqr 3654	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
44	15, 43	ssexd 4805	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ V)
45	20, 43	fssresd 6071	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶Top)
46		ptunhmeo.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (∏t‘(𝐹 ↾ 𝐵))
47	46	ptuni 21397	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ V ∧ (𝐹 ↾ 𝐵):𝐵⟶Top) → X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ 𝐿)
48	44, 45, 47	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → X𝑘 ∈ 𝐵 ∪ ((𝐹 ↾ 𝐵)‘𝑘) = ∪ 𝐿)
49	41, 48	syl5eqr 2670	. . . . . . . . 9 ⊢ (𝜑 → X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘) = ∪ 𝐿)
50		ptunhmeo.y	. . . . . . . . 9 ⊢ 𝑌 = ∪ 𝐿
51	49, 50	syl6eqr 2674	. . . . . . . 8 ⊢ (𝜑 → X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘) = 𝑌)
52	37, 51	eqtrd 2656	. . . . . . 7 ⊢ (𝜑 → X𝑘 ∈ (𝐶 ∖ 𝐴)∪ (𝐹‘𝑘) = 𝑌)
53	52	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑋 × 𝑌)) → X𝑘 ∈ (𝐶 ∖ 𝐴)∪ (𝐹‘𝑘) = 𝑌)
54	31, 53	eleqtrrd 2704	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑋 × 𝑌)) → (2nd ‘𝑤) ∈ X𝑘 ∈ (𝐶 ∖ 𝐴)∪ (𝐹‘𝑘))
55	18	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑋 × 𝑌)) → 𝐴 ⊆ 𝐶)
56		undifixp 7944	. . . . 5 ⊢ (((1st ‘𝑤) ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∧ (2nd ‘𝑤) ∈ X𝑘 ∈ (𝐶 ∖ 𝐴)∪ (𝐹‘𝑘) ∧ 𝐴 ⊆ 𝐶) → ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∈ X𝑘 ∈ 𝐶 ∪ (𝐹‘𝑘))
57	29, 54, 55, 56	syl3anc 1326	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑋 × 𝑌)) → ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∈ X𝑘 ∈ 𝐶 ∪ (𝐹‘𝑘))
58		ptunhmeo.j	. . . . . . 7 ⊢ 𝐽 = (∏t‘𝐹)
59	58	ptuni 21397	. . . . . 6 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐹:𝐶⟶Top) → X𝑘 ∈ 𝐶 ∪ (𝐹‘𝑘) = ∪ 𝐽)
60	15, 20, 59	syl2anc 693	. . . . 5 ⊢ (𝜑 → X𝑘 ∈ 𝐶 ∪ (𝐹‘𝑘) = ∪ 𝐽)
61	60	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑋 × 𝑌)) → X𝑘 ∈ 𝐶 ∪ (𝐹‘𝑘) = ∪ 𝐽)
62	57, 61	eleqtrd 2703	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑋 × 𝑌)) → ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ∈ ∪ 𝐽)
63	18	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝐽) → 𝐴 ⊆ 𝐶)
64	60	eleq2d 2687	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ X𝑘 ∈ 𝐶 ∪ (𝐹‘𝑘) ↔ 𝑧 ∈ ∪ 𝐽))
65	64	biimpar 502	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝐽) → 𝑧 ∈ X𝑘 ∈ 𝐶 ∪ (𝐹‘𝑘))
66		resixp 7943	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝑧 ∈ X𝑘 ∈ 𝐶 ∪ (𝐹‘𝑘)) → (𝑧 ↾ 𝐴) ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
67	63, 65, 66	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝐽) → (𝑧 ↾ 𝐴) ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘))
68	27	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝐽) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = 𝑋)
69	67, 68	eleqtrd 2703	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝐽) → (𝑧 ↾ 𝐴) ∈ 𝑋)
70	43	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝐽) → 𝐵 ⊆ 𝐶)
71		resixp 7943	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝑧 ∈ X𝑘 ∈ 𝐶 ∪ (𝐹‘𝑘)) → (𝑧 ↾ 𝐵) ∈ X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘))
72	70, 65, 71	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝐽) → (𝑧 ↾ 𝐵) ∈ X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘))
73	51	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝐽) → X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘) = 𝑌)
74	72, 73	eleqtrd 2703	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝐽) → (𝑧 ↾ 𝐵) ∈ 𝑌)
75		opelxpi 5148	. . . 4 ⊢ (((𝑧 ↾ 𝐴) ∈ 𝑋 ∧ (𝑧 ↾ 𝐵) ∈ 𝑌) → ⟨(𝑧 ↾ 𝐴), (𝑧 ↾ 𝐵)⟩ ∈ (𝑋 × 𝑌))
76	69, 74, 75	syl2anc 693	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ 𝐽) → ⟨(𝑧 ↾ 𝐴), (𝑧 ↾ 𝐵)⟩ ∈ (𝑋 × 𝑌))
77		eqop 7208	. . . . 5 ⊢ (𝑤 ∈ (𝑋 × 𝑌) → (𝑤 = ⟨(𝑧 ↾ 𝐴), (𝑧 ↾ 𝐵)⟩ ↔ ((1st ‘𝑤) = (𝑧 ↾ 𝐴) ∧ (2nd ‘𝑤) = (𝑧 ↾ 𝐵))))
78	77	ad2antrl 764	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (𝑤 = ⟨(𝑧 ↾ 𝐴), (𝑧 ↾ 𝐵)⟩ ↔ ((1st ‘𝑤) = (𝑧 ↾ 𝐴) ∧ (2nd ‘𝑤) = (𝑧 ↾ 𝐵))))
79	65	adantrl 752	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → 𝑧 ∈ X𝑘 ∈ 𝐶 ∪ (𝐹‘𝑘))
80		ixpfn 7914	. . . . . . . . 9 ⊢ (𝑧 ∈ X𝑘 ∈ 𝐶 ∪ (𝐹‘𝑘) → 𝑧 Fn 𝐶)
81		fnresdm 6000	. . . . . . . . 9 ⊢ (𝑧 Fn 𝐶 → (𝑧 ↾ 𝐶) = 𝑧)
82	79, 80, 81	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (𝑧 ↾ 𝐶) = 𝑧)
83	17	reseq2d 5396	. . . . . . . . 9 ⊢ (𝜑 → (𝑧 ↾ 𝐶) = (𝑧 ↾ (𝐴 ∪ 𝐵)))
84	83	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (𝑧 ↾ 𝐶) = (𝑧 ↾ (𝐴 ∪ 𝐵)))
85	82, 84	eqtr3d 2658	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → 𝑧 = (𝑧 ↾ (𝐴 ∪ 𝐵)))
86		resundi 5410	. . . . . . 7 ⊢ (𝑧 ↾ (𝐴 ∪ 𝐵)) = ((𝑧 ↾ 𝐴) ∪ (𝑧 ↾ 𝐵))
87	85, 86	syl6eq 2672	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → 𝑧 = ((𝑧 ↾ 𝐴) ∪ (𝑧 ↾ 𝐵)))
88		uneq12 3762	. . . . . . 7 ⊢ (((1st ‘𝑤) = (𝑧 ↾ 𝐴) ∧ (2nd ‘𝑤) = (𝑧 ↾ 𝐵)) → ((1st ‘𝑤) ∪ (2nd ‘𝑤)) = ((𝑧 ↾ 𝐴) ∪ (𝑧 ↾ 𝐵)))
89	88	eqeq2d 2632	. . . . . 6 ⊢ (((1st ‘𝑤) = (𝑧 ↾ 𝐴) ∧ (2nd ‘𝑤) = (𝑧 ↾ 𝐵)) → (𝑧 = ((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↔ 𝑧 = ((𝑧 ↾ 𝐴) ∪ (𝑧 ↾ 𝐵))))
90	87, 89	syl5ibrcom 237	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (((1st ‘𝑤) = (𝑧 ↾ 𝐴) ∧ (2nd ‘𝑤) = (𝑧 ↾ 𝐵)) → 𝑧 = ((1st ‘𝑤) ∪ (2nd ‘𝑤))))
91		ixpfn 7914	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑤) ∈ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) → (1st ‘𝑤) Fn 𝐴)
92	29, 91	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑋 × 𝑌)) → (1st ‘𝑤) Fn 𝐴)
93	92	adantrr 753	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (1st ‘𝑤) Fn 𝐴)
94		dffn2 6047	. . . . . . . . . 10 ⊢ ((1st ‘𝑤) Fn 𝐴 ↔ (1st ‘𝑤):𝐴⟶V)
95	93, 94	sylib 208	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (1st ‘𝑤):𝐴⟶V)
96	51	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑋 × 𝑌)) → X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘) = 𝑌)
97	31, 96	eleqtrrd 2704	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑋 × 𝑌)) → (2nd ‘𝑤) ∈ X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘))
98		ixpfn 7914	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑤) ∈ X𝑘 ∈ 𝐵 ∪ (𝐹‘𝑘) → (2nd ‘𝑤) Fn 𝐵)
99	97, 98	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝑋 × 𝑌)) → (2nd ‘𝑤) Fn 𝐵)
100	99	adantrr 753	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (2nd ‘𝑤) Fn 𝐵)
101		dffn2 6047	. . . . . . . . . 10 ⊢ ((2nd ‘𝑤) Fn 𝐵 ↔ (2nd ‘𝑤):𝐵⟶V)
102	100, 101	sylib 208	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (2nd ‘𝑤):𝐵⟶V)
103		res0 5400	. . . . . . . . . . 11 ⊢ ((1st ‘𝑤) ↾ ∅) = ∅
104		res0 5400	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑤) ↾ ∅) = ∅
105	103, 104	eqtr4i 2647	. . . . . . . . . 10 ⊢ ((1st ‘𝑤) ↾ ∅) = ((2nd ‘𝑤) ↾ ∅)
106	33	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (𝐴 ∩ 𝐵) = ∅)
107	106	reseq2d 5396	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → ((1st ‘𝑤) ↾ (𝐴 ∩ 𝐵)) = ((1st ‘𝑤) ↾ ∅))
108	106	reseq2d 5396	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → ((2nd ‘𝑤) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑤) ↾ ∅))
109	105, 107, 108	3eqtr4a 2682	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → ((1st ‘𝑤) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑤) ↾ (𝐴 ∩ 𝐵)))
110		fresaunres1 6077	. . . . . . . . 9 ⊢ (((1st ‘𝑤):𝐴⟶V ∧ (2nd ‘𝑤):𝐵⟶V ∧ ((1st ‘𝑤) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑤) ↾ (𝐴 ∩ 𝐵))) → (((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↾ 𝐴) = (1st ‘𝑤))
111	95, 102, 109, 110	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↾ 𝐴) = (1st ‘𝑤))
112	111	eqcomd 2628	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (1st ‘𝑤) = (((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↾ 𝐴))
113		fresaunres2 6076	. . . . . . . . 9 ⊢ (((1st ‘𝑤):𝐴⟶V ∧ (2nd ‘𝑤):𝐵⟶V ∧ ((1st ‘𝑤) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑤) ↾ (𝐴 ∩ 𝐵))) → (((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↾ 𝐵) = (2nd ‘𝑤))
114	95, 102, 109, 113	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↾ 𝐵) = (2nd ‘𝑤))
115	114	eqcomd 2628	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (2nd ‘𝑤) = (((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↾ 𝐵))
116	112, 115	jca 554	. . . . . 6 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → ((1st ‘𝑤) = (((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↾ 𝐴) ∧ (2nd ‘𝑤) = (((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↾ 𝐵)))
117		reseq1 5390	. . . . . . . 8 ⊢ (𝑧 = ((1st ‘𝑤) ∪ (2nd ‘𝑤)) → (𝑧 ↾ 𝐴) = (((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↾ 𝐴))
118	117	eqeq2d 2632	. . . . . . 7 ⊢ (𝑧 = ((1st ‘𝑤) ∪ (2nd ‘𝑤)) → ((1st ‘𝑤) = (𝑧 ↾ 𝐴) ↔ (1st ‘𝑤) = (((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↾ 𝐴)))
119		reseq1 5390	. . . . . . . 8 ⊢ (𝑧 = ((1st ‘𝑤) ∪ (2nd ‘𝑤)) → (𝑧 ↾ 𝐵) = (((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↾ 𝐵))
120	119	eqeq2d 2632	. . . . . . 7 ⊢ (𝑧 = ((1st ‘𝑤) ∪ (2nd ‘𝑤)) → ((2nd ‘𝑤) = (𝑧 ↾ 𝐵) ↔ (2nd ‘𝑤) = (((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↾ 𝐵)))
121	118, 120	anbi12d 747	. . . . . 6 ⊢ (𝑧 = ((1st ‘𝑤) ∪ (2nd ‘𝑤)) → (((1st ‘𝑤) = (𝑧 ↾ 𝐴) ∧ (2nd ‘𝑤) = (𝑧 ↾ 𝐵)) ↔ ((1st ‘𝑤) = (((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↾ 𝐴) ∧ (2nd ‘𝑤) = (((1st ‘𝑤) ∪ (2nd ‘𝑤)) ↾ 𝐵))))
122	116, 121	syl5ibrcom 237	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (𝑧 = ((1st ‘𝑤) ∪ (2nd ‘𝑤)) → ((1st ‘𝑤) = (𝑧 ↾ 𝐴) ∧ (2nd ‘𝑤) = (𝑧 ↾ 𝐵))))
123	90, 122	impbid 202	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (((1st ‘𝑤) = (𝑧 ↾ 𝐴) ∧ (2nd ‘𝑤) = (𝑧 ↾ 𝐵)) ↔ 𝑧 = ((1st ‘𝑤) ∪ (2nd ‘𝑤))))
124	78, 123	bitrd 268	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ ∪ 𝐽)) → (𝑤 = ⟨(𝑧 ↾ 𝐴), (𝑧 ↾ 𝐵)⟩ ↔ 𝑧 = ((1st ‘𝑤) ∪ (2nd ‘𝑤))))
125	8, 62, 76, 124	f1ocnv2d 6886	. 2 ⊢ (𝜑 → (𝐺:(𝑋 × 𝑌)–1-1-onto→∪ 𝐽 ∧ ◡𝐺 = (𝑧 ∈ ∪ 𝐽 ↦ ⟨(𝑧 ↾ 𝐴), (𝑧 ↾ 𝐵)⟩)))
126	125	simprd 479	1 ⊢ (𝜑 → ◡𝐺 = (𝑧 ∈ ∪ 𝐽 ↦ ⟨(𝑧 ↾ 𝐴), (𝑧 ↾ 𝐵)⟩))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ⟨cop 4183  ∪ cuni 4436   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113   ↾ cres 5116   Fn wfn 5883  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167  Xcixp 7908  ∏_tcpt 16099  Topctop 20698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-ixp 7909  df-en 7956  df-fin 7959  df-fi 8317  df-topgen 16104  df-pt 16105  df-top 20699  df-bases 20750

This theorem is referenced by:  ptunhmeo  21611