Theorem cnmpt2t 23681

Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
cnmpt21.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmpt21.a	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
cnmpt2t.b	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))

Assertion

Ref	Expression
cnmpt2t	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))

Distinct variable groups:   𝑥,𝑦,𝐿   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑀,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt2t

Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6906	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑢, 𝑣⟩))
2		df-ov 7434	. . . . . . 7 ⊢ (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑢, 𝑣⟩)
3	1, 2	eqtr4di 2795	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧) = (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣))
4		fveq2 6906	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘⟨𝑢, 𝑣⟩))
5		df-ov 7434	. . . . . . 7 ⊢ (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘⟨𝑢, 𝑣⟩)
6	4, 5	eqtr4di 2795	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧) = (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣))
7	3, 6	opeq12d 4881	. . . . 5 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ⟨((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧), ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧)⟩ = ⟨(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣), (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)⟩)
8	7	mpompt 7547	. . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧), ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧)⟩) = (𝑢 ∈ 𝑋, 𝑣 ∈ 𝑌 ↦ ⟨(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣), (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)⟩)
9		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑥𝑢
10		nfmpo1 7513	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
11		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑥𝑣
12	9, 10, 11	nfov 7461	. . . . . 6 ⊢ Ⅎ𝑥(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣)
13		nfmpo1 7513	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)
14	9, 13, 11	nfov 7461	. . . . . 6 ⊢ Ⅎ𝑥(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)
15	12, 14	nfop 4889	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣), (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)⟩
16		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑦𝑢
17		nfmpo2 7514	. . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
18		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
19	16, 17, 18	nfov 7461	. . . . . 6 ⊢ Ⅎ𝑦(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣)
20		nfmpo2 7514	. . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)
21	16, 20, 18	nfov 7461	. . . . . 6 ⊢ Ⅎ𝑦(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)
22	19, 21	nfop 4889	. . . . 5 ⊢ Ⅎ𝑦⟨(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣), (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)⟩
23		nfcv 2905	. . . . 5 ⊢ Ⅎ𝑢⟨(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦), (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦)⟩
24		nfcv 2905	. . . . 5 ⊢ Ⅎ𝑣⟨(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦), (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦)⟩
25		oveq12 7440	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣) = (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦))
26		oveq12 7440	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣) = (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦))
27	25, 26	opeq12d 4881	. . . . 5 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → ⟨(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣), (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)⟩ = ⟨(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦), (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦)⟩)
28	15, 22, 23, 24, 27	cbvmpo 7527	. . . 4 ⊢ (𝑢 ∈ 𝑋, 𝑣 ∈ 𝑌 ↦ ⟨(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣), (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦), (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦)⟩)
29	8, 28	eqtri 2765	. . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧), ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧)⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦), (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦)⟩)
30		cnmpt21.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
31		cnmpt21.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
32		txtopon 23599	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
33	30, 31, 32	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
34		toponuni 22920	. . . 4 ⊢ ((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
35		mpteq1 5235	. . . 4 ⊢ ((𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾) → (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧), ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧)⟩) = (𝑧 ∈ ∪ (𝐽 ×t 𝐾) ↦ ⟨((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧), ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧)⟩))
36	33, 34, 35	3syl 18	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧), ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧)⟩) = (𝑧 ∈ ∪ (𝐽 ×t 𝐾) ↦ ⟨((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧), ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧)⟩))
37		simp2 1138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑥 ∈ 𝑋)
38		simp3 1139	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
39		cnmpt21.a	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
40		cntop2 23249	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
41	39, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ Top)
42		toptopon2 22924	. . . . . . . . . . 11 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
43	41, 42	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
44		cnf2 23257	. . . . . . . . . 10 ⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶∪ 𝐿)
45	33, 43, 39, 44	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶∪ 𝐿)
46		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
47	46	fmpo 8093	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶∪ 𝐿)
48	45, 47	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿)
49		rsp2 3277	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ ∪ 𝐿))
50	48, 49	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ ∪ 𝐿))
51	50	3impib 1117	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ ∪ 𝐿)
52	46	ovmpt4g 7580	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
53	37, 38, 51, 52	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
54		cnmpt2t.b	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
55		cntop2 23249	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
56	54, 55	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ Top)
57		toptopon2 22924	. . . . . . . . . . 11 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
58	56, 57	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
59		cnf2 23257	. . . . . . . . . 10 ⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑀 ∈ (TopOn‘∪ 𝑀) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶∪ 𝑀)
60	33, 58, 54, 59	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶∪ 𝑀)
61		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)
62	61	fmpo 8093	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ ∪ 𝑀 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶∪ 𝑀)
63	60, 62	sylibr 234	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ ∪ 𝑀)
64		rsp2 3277	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ ∪ 𝑀 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ ∪ 𝑀))
65	63, 64	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ ∪ 𝑀))
66	65	3impib 1117	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ ∪ 𝑀)
67	61	ovmpt4g 7580	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐵 ∈ ∪ 𝑀) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦) = 𝐵)
68	37, 38, 66, 67	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦) = 𝐵)
69	53, 68	opeq12d 4881	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦), (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦)⟩ = ⟨𝐴, 𝐵⟩)
70	69	mpoeq3dva 7510	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦), (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦)⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝐵⟩))
71	29, 36, 70	3eqtr3a 2801	. 2 ⊢ (𝜑 → (𝑧 ∈ ∪ (𝐽 ×t 𝐾) ↦ ⟨((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧), ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧)⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝐵⟩))
72		eqid 2737	. . . 4 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
73		eqid 2737	. . . 4 ⊢ (𝑧 ∈ ∪ (𝐽 ×t 𝐾) ↦ ⟨((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧), ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧)⟩) = (𝑧 ∈ ∪ (𝐽 ×t 𝐾) ↦ ⟨((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧), ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧)⟩)
74	72, 73	txcnmpt 23632	. . 3 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) → (𝑧 ∈ ∪ (𝐽 ×t 𝐾) ↦ ⟨((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧), ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧)⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
75	39, 54, 74	syl2anc 584	. 2 ⊢ (𝜑 → (𝑧 ∈ ∪ (𝐽 ×t 𝐾) ↦ ⟨((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧), ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧)⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
76	71, 75	eqeltrrd 2842	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ⟨cop 4632  ∪ cuni 4907   ↦ cmpt 5225   × cxp 5683  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  Topctop 22899  TopOnctopon 22916   Cn ccn 23232   ×_t ctx 23568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cn 23235  df-tx 23570

This theorem is referenced by:  cnmpt22  23682  txhmeo  23811  txswaphmeo  23813  txsconnlem  35245