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Theorem txhmeo 21416

Description: Lift a pair of homeomorphisms on the factors to a homeomorphism of product topologies. (Contributed by Mario Carneiro, 2-Sep-2015.)

**Hypotheses**
Ref	Expression
txhmeo.1	⊢ 𝑋 = ∪ 𝐽
txhmeo.2	⊢ 𝑌 = ∪ 𝐾
txhmeo.3	⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo𝐿))
txhmeo.4	⊢ (𝜑 → 𝐺 ∈ (𝐾Homeo𝑀))

**Assertion**
Ref	Expression
txhmeo	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×_t 𝐾)Homeo(𝐿 ×_t 𝑀)))

Distinct variable groups: 𝑥,𝑦,𝐹 𝑥,𝐽,𝑦 𝑥,𝐾,𝑦 𝜑,𝑥,𝑦 𝑥,𝐺,𝑦 𝑥,𝐿,𝑦 𝑥,𝑋,𝑦 𝑥,𝑌,𝑦 𝑥,𝑀,𝑦

Proof of Theorem txhmeo Dummy variables 𝑣 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		txhmeo.3	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐽Homeo𝐿))
2		hmeocn 21373	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐿) → 𝐹 ∈ (𝐽 Cn 𝐿))
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐿))
4		cntop1 20854	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐿) → 𝐽 ∈ Top)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐽 ∈ Top)
6		txhmeo.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
7	6	toptopon 20548	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
8	5, 7	sylib 207	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
9		txhmeo.4	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (𝐾Homeo𝑀))
10		hmeocn 21373	. . . . . 6 ⊢ (𝐺 ∈ (𝐾Homeo𝑀) → 𝐺 ∈ (𝐾 Cn 𝑀))
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝑀))
12		cntop1 20854	. . . . 5 ⊢ (𝐺 ∈ (𝐾 Cn 𝑀) → 𝐾 ∈ Top)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Top)
14		txhmeo.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
15	14	toptopon 20548	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
16	13, 15	sylib 207	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
17	8, 16	cnmpt1st 21281	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×_t 𝐾) Cn 𝐽))
18	8, 16, 17, 3	cnmpt21f 21285	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐹‘𝑥)) ∈ ((𝐽 ×_t 𝐾) Cn 𝐿))
19	8, 16	cnmpt2nd 21282	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×_t 𝐾) Cn 𝐾))
20	8, 16, 19, 11	cnmpt21f 21285	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐺‘𝑦)) ∈ ((𝐽 ×_t 𝐾) Cn 𝑀))
21	8, 16, 18, 20	cnmpt2t 21286	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×_t 𝐾) Cn (𝐿 ×_t 𝑀)))
22		vex 3176	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
23		vex 3176	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
24	22, 23	op1std 7069	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑢) = 𝑥)
25	24	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1^st ‘𝑢)) = (𝐹‘𝑥))
26	22, 23	op2ndd 7070	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (2^nd ‘𝑢) = 𝑦)
27	26	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → (𝐺‘(2^nd ‘𝑢)) = (𝐺‘𝑦))
28	25, 27	opeq12d 4348	. . . . . . . 8 ⊢ (𝑢 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩ = ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
29	28	mpt2mpt 6650	. . . . . . 7 ⊢ (𝑢 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)
30	29	eqcomi 2619	. . . . . 6 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑢 ∈ (𝑋 × 𝑌) ↦ ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩)
31		eqid 2610	. . . . . . . . . 10 ⊢ ∪ 𝐿 = ∪ 𝐿
32	6, 31	cnf 20860	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝐽 Cn 𝐿) → 𝐹:𝑋⟶∪ 𝐿)
33	3, 32	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝐿)
34		xp1st 7089	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (1^st ‘𝑢) ∈ 𝑋)
35		ffvelrn 6265	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶∪ 𝐿 ∧ (1^st ‘𝑢) ∈ 𝑋) → (𝐹‘(1^st ‘𝑢)) ∈ ∪ 𝐿)
36	33, 34, 35	syl2an 493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑋 × 𝑌)) → (𝐹‘(1^st ‘𝑢)) ∈ ∪ 𝐿)
37		eqid 2610	. . . . . . . . . 10 ⊢ ∪ 𝑀 = ∪ 𝑀
38	14, 37	cnf 20860	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝐾 Cn 𝑀) → 𝐺:𝑌⟶∪ 𝑀)
39	11, 38	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑌⟶∪ 𝑀)
40		xp2nd 7090	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑢) ∈ 𝑌)
41		ffvelrn 6265	. . . . . . . 8 ⊢ ((𝐺:𝑌⟶∪ 𝑀 ∧ (2^nd ‘𝑢) ∈ 𝑌) → (𝐺‘(2^nd ‘𝑢)) ∈ ∪ 𝑀)
42	39, 40, 41	syl2an 493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑋 × 𝑌)) → (𝐺‘(2^nd ‘𝑢)) ∈ ∪ 𝑀)
43		opelxpi 5072	. . . . . . 7 ⊢ (((𝐹‘(1^st ‘𝑢)) ∈ ∪ 𝐿 ∧ (𝐺‘(2^nd ‘𝑢)) ∈ ∪ 𝑀) → ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩ ∈ (∪ 𝐿 × ∪ 𝑀))
44	36, 42, 43	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝑋 × 𝑌)) → ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩ ∈ (∪ 𝐿 × ∪ 𝑀))
45	6, 31	hmeof1o 21377	. . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐽Homeo𝐿) → 𝐹:𝑋–1-1-onto→∪ 𝐿)
46	1, 45	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→∪ 𝐿)
47		f1ocnv 6062	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐿 → ^◡𝐹:∪ 𝐿–1-1-onto→𝑋)
48		f1of 6050	. . . . . . . . 9 ⊢ (^◡𝐹:∪ 𝐿–1-1-onto→𝑋 → ^◡𝐹:∪ 𝐿⟶𝑋)
49	46, 47, 48	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ^◡𝐹:∪ 𝐿⟶𝑋)
50		xp1st 7089	. . . . . . . 8 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) → (1^st ‘𝑣) ∈ ∪ 𝐿)
51		ffvelrn 6265	. . . . . . . 8 ⊢ ((^◡𝐹:∪ 𝐿⟶𝑋 ∧ (1^st ‘𝑣) ∈ ∪ 𝐿) → (^◡𝐹‘(1^st ‘𝑣)) ∈ 𝑋)
52	49, 50, 51	syl2an 493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀)) → (^◡𝐹‘(1^st ‘𝑣)) ∈ 𝑋)
53	14, 37	hmeof1o 21377	. . . . . . . . . 10 ⊢ (𝐺 ∈ (𝐾Homeo𝑀) → 𝐺:𝑌–1-1-onto→∪ 𝑀)
54	9, 53	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝑌–1-1-onto→∪ 𝑀)
55		f1ocnv 6062	. . . . . . . . 9 ⊢ (𝐺:𝑌–1-1-onto→∪ 𝑀 → ^◡𝐺:∪ 𝑀–1-1-onto→𝑌)
56		f1of 6050	. . . . . . . . 9 ⊢ (^◡𝐺:∪ 𝑀–1-1-onto→𝑌 → ^◡𝐺:∪ 𝑀⟶𝑌)
57	54, 55, 56	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ^◡𝐺:∪ 𝑀⟶𝑌)
58		xp2nd 7090	. . . . . . . 8 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) → (2^nd ‘𝑣) ∈ ∪ 𝑀)
59		ffvelrn 6265	. . . . . . . 8 ⊢ ((^◡𝐺:∪ 𝑀⟶𝑌 ∧ (2^nd ‘𝑣) ∈ ∪ 𝑀) → (^◡𝐺‘(2^nd ‘𝑣)) ∈ 𝑌)
60	57, 58, 59	syl2an 493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀)) → (^◡𝐺‘(2^nd ‘𝑣)) ∈ 𝑌)
61		opelxpi 5072	. . . . . . 7 ⊢ (((^◡𝐹‘(1^st ‘𝑣)) ∈ 𝑋 ∧ (^◡𝐺‘(2^nd ‘𝑣)) ∈ 𝑌) → ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩ ∈ (𝑋 × 𝑌))
62	52, 60, 61	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀)) → ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩ ∈ (𝑋 × 𝑌))
63	46	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → 𝐹:𝑋–1-1-onto→∪ 𝐿)
64	34	ad2antrl 760	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (1^st ‘𝑢) ∈ 𝑋)
65	50	ad2antll 761	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (1^st ‘𝑣) ∈ ∪ 𝐿)
66		f1ocnvfvb 6435	. . . . . . . . . 10 ⊢ ((𝐹:𝑋–1-1-onto→∪ 𝐿 ∧ (1^st ‘𝑢) ∈ 𝑋 ∧ (1^st ‘𝑣) ∈ ∪ 𝐿) → ((𝐹‘(1^st ‘𝑢)) = (1^st ‘𝑣) ↔ (^◡𝐹‘(1^st ‘𝑣)) = (1^st ‘𝑢)))
67	63, 64, 65, 66	syl3anc 1318	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((𝐹‘(1^st ‘𝑢)) = (1^st ‘𝑣) ↔ (^◡𝐹‘(1^st ‘𝑣)) = (1^st ‘𝑢)))
68		eqcom 2617	. . . . . . . . 9 ⊢ ((1^st ‘𝑣) = (𝐹‘(1^st ‘𝑢)) ↔ (𝐹‘(1^st ‘𝑢)) = (1^st ‘𝑣))
69		eqcom 2617	. . . . . . . . 9 ⊢ ((1^st ‘𝑢) = (^◡𝐹‘(1^st ‘𝑣)) ↔ (^◡𝐹‘(1^st ‘𝑣)) = (1^st ‘𝑢))
70	67, 68, 69	3bitr4g 302	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((1^st ‘𝑣) = (𝐹‘(1^st ‘𝑢)) ↔ (1^st ‘𝑢) = (^◡𝐹‘(1^st ‘𝑣))))
71	54	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → 𝐺:𝑌–1-1-onto→∪ 𝑀)
72	40	ad2antrl 760	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (2^nd ‘𝑢) ∈ 𝑌)
73	58	ad2antll 761	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (2^nd ‘𝑣) ∈ ∪ 𝑀)
74		f1ocnvfvb 6435	. . . . . . . . . 10 ⊢ ((𝐺:𝑌–1-1-onto→∪ 𝑀 ∧ (2^nd ‘𝑢) ∈ 𝑌 ∧ (2^nd ‘𝑣) ∈ ∪ 𝑀) → ((𝐺‘(2^nd ‘𝑢)) = (2^nd ‘𝑣) ↔ (^◡𝐺‘(2^nd ‘𝑣)) = (2^nd ‘𝑢)))
75	71, 72, 73, 74	syl3anc 1318	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((𝐺‘(2^nd ‘𝑢)) = (2^nd ‘𝑣) ↔ (^◡𝐺‘(2^nd ‘𝑣)) = (2^nd ‘𝑢)))
76		eqcom 2617	. . . . . . . . 9 ⊢ ((2^nd ‘𝑣) = (𝐺‘(2^nd ‘𝑢)) ↔ (𝐺‘(2^nd ‘𝑢)) = (2^nd ‘𝑣))
77		eqcom 2617	. . . . . . . . 9 ⊢ ((2^nd ‘𝑢) = (^◡𝐺‘(2^nd ‘𝑣)) ↔ (^◡𝐺‘(2^nd ‘𝑣)) = (2^nd ‘𝑢))
78	75, 76, 77	3bitr4g 302	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → ((2^nd ‘𝑣) = (𝐺‘(2^nd ‘𝑢)) ↔ (2^nd ‘𝑢) = (^◡𝐺‘(2^nd ‘𝑣))))
79	70, 78	anbi12d 743	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (((1^st ‘𝑣) = (𝐹‘(1^st ‘𝑢)) ∧ (2^nd ‘𝑣) = (𝐺‘(2^nd ‘𝑢))) ↔ ((1^st ‘𝑢) = (^◡𝐹‘(1^st ‘𝑣)) ∧ (2^nd ‘𝑢) = (^◡𝐺‘(2^nd ‘𝑣)))))
80		eqop 7099	. . . . . . . 8 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) → (𝑣 = ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩ ↔ ((1^st ‘𝑣) = (𝐹‘(1^st ‘𝑢)) ∧ (2^nd ‘𝑣) = (𝐺‘(2^nd ‘𝑢)))))
81	80	ad2antll 761	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (𝑣 = ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩ ↔ ((1^st ‘𝑣) = (𝐹‘(1^st ‘𝑢)) ∧ (2^nd ‘𝑣) = (𝐺‘(2^nd ‘𝑢)))))
82		eqop 7099	. . . . . . . 8 ⊢ (𝑢 ∈ (𝑋 × 𝑌) → (𝑢 = ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩ ↔ ((1^st ‘𝑢) = (^◡𝐹‘(1^st ‘𝑣)) ∧ (2^nd ‘𝑢) = (^◡𝐺‘(2^nd ‘𝑣)))))
83	82	ad2antrl 760	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (𝑢 = ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩ ↔ ((1^st ‘𝑢) = (^◡𝐹‘(1^st ‘𝑣)) ∧ (2^nd ‘𝑢) = (^◡𝐺‘(2^nd ‘𝑣)))))
84	79, 81, 83	3bitr4rd 300	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ (𝑋 × 𝑌) ∧ 𝑣 ∈ (∪ 𝐿 × ∪ 𝑀))) → (𝑢 = ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩ ↔ 𝑣 = ⟨(𝐹‘(1^st ‘𝑢)), (𝐺‘(2^nd ‘𝑢))⟩))
85	30, 44, 62, 84	f1ocnv2d 6784	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩):(𝑋 × 𝑌)–1-1-onto→(∪ 𝐿 × ∪ 𝑀) ∧ ^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) ↦ ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩)))
86	85	simprd 478	. . . 4 ⊢ (𝜑 → ^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) ↦ ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩))
87		vex 3176	. . . . . . . 8 ⊢ 𝑧 ∈ V
88		vex 3176	. . . . . . . 8 ⊢ 𝑤 ∈ V
89	87, 88	op1std 7069	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (1^st ‘𝑣) = 𝑧)
90	89	fveq2d 6107	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (^◡𝐹‘(1^st ‘𝑣)) = (^◡𝐹‘𝑧))
91	87, 88	op2ndd 7070	. . . . . . 7 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (2^nd ‘𝑣) = 𝑤)
92	91	fveq2d 6107	. . . . . 6 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → (^◡𝐺‘(2^nd ‘𝑣)) = (^◡𝐺‘𝑤))
93	90, 92	opeq12d 4348	. . . . 5 ⊢ (𝑣 = ⟨𝑧, 𝑤⟩ → ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩ = ⟨(^◡𝐹‘𝑧), (^◡𝐺‘𝑤)⟩)
94	93	mpt2mpt 6650	. . . 4 ⊢ (𝑣 ∈ (∪ 𝐿 × ∪ 𝑀) ↦ ⟨(^◡𝐹‘(1^st ‘𝑣)), (^◡𝐺‘(2^nd ‘𝑣))⟩) = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ ⟨(^◡𝐹‘𝑧), (^◡𝐺‘𝑤)⟩)
95	86, 94	syl6eq 2660	. . 3 ⊢ (𝜑 → ^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) = (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ ⟨(^◡𝐹‘𝑧), (^◡𝐺‘𝑤)⟩))
96		cntop2 20855	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
97	3, 96	syl 17	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ Top)
98	31	toptopon 20548	. . . . 5 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
99	97, 98	sylib 207	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
100		cntop2 20855	. . . . . 6 ⊢ (𝐺 ∈ (𝐾 Cn 𝑀) → 𝑀 ∈ Top)
101	11, 100	syl 17	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ Top)
102	37	toptopon 20548	. . . . 5 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
103	101, 102	sylib 207	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
104	99, 103	cnmpt1st 21281	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ 𝑧) ∈ ((𝐿 ×_t 𝑀) Cn 𝐿))
105		hmeocnvcn 21374	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐿) → ^◡𝐹 ∈ (𝐿 Cn 𝐽))
106	1, 105	syl 17	. . . . 5 ⊢ (𝜑 → ^◡𝐹 ∈ (𝐿 Cn 𝐽))
107	99, 103, 104, 106	cnmpt21f 21285	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (^◡𝐹‘𝑧)) ∈ ((𝐿 ×_t 𝑀) Cn 𝐽))
108	99, 103	cnmpt2nd 21282	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ 𝑤) ∈ ((𝐿 ×_t 𝑀) Cn 𝑀))
109		hmeocnvcn 21374	. . . . . 6 ⊢ (𝐺 ∈ (𝐾Homeo𝑀) → ^◡𝐺 ∈ (𝑀 Cn 𝐾))
110	9, 109	syl 17	. . . . 5 ⊢ (𝜑 → ^◡𝐺 ∈ (𝑀 Cn 𝐾))
111	99, 103, 108, 110	cnmpt21f 21285	. . . 4 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ (^◡𝐺‘𝑤)) ∈ ((𝐿 ×_t 𝑀) Cn 𝐾))
112	99, 103, 107, 111	cnmpt2t 21286	. . 3 ⊢ (𝜑 → (𝑧 ∈ ∪ 𝐿, 𝑤 ∈ ∪ 𝑀 ↦ ⟨(^◡𝐹‘𝑧), (^◡𝐺‘𝑤)⟩) ∈ ((𝐿 ×_t 𝑀) Cn (𝐽 ×_t 𝐾)))
113	95, 112	eqeltrd 2688	. 2 ⊢ (𝜑 → ^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐿 ×_t 𝑀) Cn (𝐽 ×_t 𝐾)))
114		ishmeo 21372	. 2 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×_t 𝐾)Homeo(𝐿 ×_t 𝑀)) ↔ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×_t 𝐾) Cn (𝐿 ×_t 𝑀)) ∧ ^◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐿 ×_t 𝑀) Cn (𝐽 ×_t 𝐾))))
115	21, 113, 114	sylanbrc 695	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩) ∈ ((𝐽 ×_t 𝐾)Homeo(𝐿 ×_t 𝑀)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⟨cop 4131 ∪ cuni 4372 ↦ cmpt 4643 × cxp 5036 ^◡ccnv 5037 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1^st c1st 7057 2^nd c2nd 7058 Topctop 20517 TopOnctopon 20518 Cn ccn 20838 ×_t ctx 21173 Homeochmeo 21366

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-map 7746 df-topgen 15927 df-top 20521 df-bases 20522 df-topon 20523 df-cn 20841 df-tx 21175 df-hmeo 21368

This theorem is referenced by: xpstopnlem1 21422

W3C validator