Theorem cnmpt2t 14529

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 5558	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑢, 𝑣⟩))
2		df-ov 5925	. . . . . . 7 ⊢ (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑢, 𝑣⟩)
3	1, 2	eqtr4di 2247	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧) = (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣))
4		fveq2 5558	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘⟨𝑢, 𝑣⟩))
5		df-ov 5925	. . . . . . 7 ⊢ (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘⟨𝑢, 𝑣⟩)
6	4, 5	eqtr4di 2247	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧) = (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣))
7	3, 6	opeq12d 3816	. . . . 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 6014	. . . 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 2339	. . . . . . 7 ⊢ Ⅎ𝑥𝑢
10		nfmpo1 5989	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
11		nfcv 2339	. . . . . . 7 ⊢ Ⅎ𝑥𝑣
12	9, 10, 11	nfov 5952	. . . . . 6 ⊢ Ⅎ𝑥(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣)
13		nfmpo1 5989	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)
14	9, 13, 11	nfov 5952	. . . . . 6 ⊢ Ⅎ𝑥(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)
15	12, 14	nfop 3824	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣), (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)⟩
16		nfcv 2339	. . . . . . 7 ⊢ Ⅎ𝑦𝑢
17		nfmpo2 5990	. . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
18		nfcv 2339	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
19	16, 17, 18	nfov 5952	. . . . . 6 ⊢ Ⅎ𝑦(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣)
20		nfmpo2 5990	. . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)
21	16, 20, 18	nfov 5952	. . . . . 6 ⊢ Ⅎ𝑦(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)
22	19, 21	nfop 3824	. . . . 5 ⊢ Ⅎ𝑦⟨(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣), (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)⟩
23		nfcv 2339	. . . . 5 ⊢ Ⅎ𝑢⟨(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦), (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦)⟩
24		nfcv 2339	. . . . 5 ⊢ Ⅎ𝑣⟨(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦), (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦)⟩
25		oveq12 5931	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣) = (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦))
26		oveq12 5931	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣) = (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦))
27	25, 26	opeq12d 3816	. . . . 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 6001	. . . 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 2217	. . 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 14498	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
33	30, 31, 32	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
34		toponuni 14251	. . . 4 ⊢ ((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
35		mpteq1 4117	. . . 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 17	. . 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 1000	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑥 ∈ 𝑋)
38		simp3 1001	. . . . . 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 14438	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
41	39, 40	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ Top)
42		toptopon2 14255	. . . . . . . . . . 11 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
43	41, 42	sylib 122	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
44		cnf2 14441	. . . . . . . . . 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 1249	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶∪ 𝐿)
46		eqid 2196	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
47	46	fmpo 6259	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶∪ 𝐿)
48	45, 47	sylibr 134	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿)
49		rsp2 2547	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ ∪ 𝐿))
50	48, 49	syl 14	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ ∪ 𝐿))
51	50	3impib 1203	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ ∪ 𝐿)
52	46	ovmpt4g 6045	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
53	37, 38, 51, 52	syl3anc 1249	. . . . 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 14438	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
56	54, 55	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ Top)
57		toptopon2 14255	. . . . . . . . . . 11 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
58	56, 57	sylib 122	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
59		cnf2 14441	. . . . . . . . . 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 1249	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶∪ 𝑀)
61		eqid 2196	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)
62	61	fmpo 6259	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ ∪ 𝑀 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶∪ 𝑀)
63	60, 62	sylibr 134	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ ∪ 𝑀)
64		rsp2 2547	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ ∪ 𝑀 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ ∪ 𝑀))
65	63, 64	syl 14	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ ∪ 𝑀))
66	65	3impib 1203	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ ∪ 𝑀)
67	61	ovmpt4g 6045	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐵 ∈ ∪ 𝑀) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦) = 𝐵)
68	37, 38, 66, 67	syl3anc 1249	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦) = 𝐵)
69	53, 68	opeq12d 3816	. . . 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 5986	. . 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 2253	. 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 2196	. . . 4 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
73		eqid 2196	. . . 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 14509	. . 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 411	. 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 2274	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ⟨cop 3625  ∪ cuni 3839   ↦ cmpt 4094   × cxp 4661  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922   ∈ cmpo 5924  Topctop 14233  TopOnctopon 14246   Cn ccn 14421   ×_t ctx 14488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-map 6709  df-topgen 12931  df-top 14234  df-topon 14247  df-bases 14279  df-cn 14424  df-tx 14489

This theorem is referenced by:  cnmpt22  14530  txhmeo  14555  txswaphmeo  14557