Theorem cnmpt2t 14736

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 5575	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑢, 𝑣⟩))
2		df-ov 5946	. . . . . . 7 ⊢ (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘⟨𝑢, 𝑣⟩)
3	1, 2	eqtr4di 2255	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)‘𝑧) = (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣))
4		fveq2 5575	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘⟨𝑢, 𝑣⟩))
5		df-ov 5946	. . . . . . 7 ⊢ (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘⟨𝑢, 𝑣⟩)
6	4, 5	eqtr4di 2255	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)‘𝑧) = (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣))
7	3, 6	opeq12d 3826	. . . . 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 6036	. . . 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 2347	. . . . . . 7 ⊢ Ⅎ𝑥𝑢
10		nfmpo1 6011	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
11		nfcv 2347	. . . . . . 7 ⊢ Ⅎ𝑥𝑣
12	9, 10, 11	nfov 5973	. . . . . 6 ⊢ Ⅎ𝑥(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣)
13		nfmpo1 6011	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)
14	9, 13, 11	nfov 5973	. . . . . 6 ⊢ Ⅎ𝑥(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)
15	12, 14	nfop 3834	. . . . 5 ⊢ Ⅎ𝑥⟨(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣), (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)⟩
16		nfcv 2347	. . . . . . 7 ⊢ Ⅎ𝑦𝑢
17		nfmpo2 6012	. . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
18		nfcv 2347	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
19	16, 17, 18	nfov 5973	. . . . . 6 ⊢ Ⅎ𝑦(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣)
20		nfmpo2 6012	. . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)
21	16, 20, 18	nfov 5973	. . . . . 6 ⊢ Ⅎ𝑦(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)
22	19, 21	nfop 3834	. . . . 5 ⊢ Ⅎ𝑦⟨(𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣), (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣)⟩
23		nfcv 2347	. . . . 5 ⊢ Ⅎ𝑢⟨(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦), (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦)⟩
24		nfcv 2347	. . . . 5 ⊢ Ⅎ𝑣⟨(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦), (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦)⟩
25		oveq12 5952	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑣) = (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦))
26		oveq12 5952	. . . . . 6 ⊢ ((𝑢 = 𝑥 ∧ 𝑣 = 𝑦) → (𝑢(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑣) = (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)𝑦))
27	25, 26	opeq12d 3826	. . . . 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 6023	. . . 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 2225	. . 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 14705	. . . . 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 14458	. . . 4 ⊢ ((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
35		mpteq1 4127	. . . 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 14645	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
41	39, 40	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ Top)
42		toptopon2 14462	. . . . . . . . . . 11 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
43	41, 42	sylib 122	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
44		cnf2 14648	. . . . . . . . . 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 2204	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
47	46	fmpo 6286	. . . . . . . . 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 2555	. . . . . . . 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 6067	. . . . . 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 14645	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
56	54, 55	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ Top)
57		toptopon2 14462	. . . . . . . . . . 11 ⊢ (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘∪ 𝑀))
58	56, 57	sylib 122	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘∪ 𝑀))
59		cnf2 14648	. . . . . . . . . 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 2204	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)
62	61	fmpo 6286	. . . . . . . . 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 2555	. . . . . . . 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 6067	. . . . . 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 3826	. . . 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 6008	. . 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 2261	. 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 2204	. . . 4 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
73		eqid 2204	. . . 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 14716	. . 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 2282	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1372   ∈ wcel 2175  ∀wral 2483  ⟨cop 3635  ∪ cuni 3849   ↦ cmpt 4104   × cxp 4672  ⟶wf 5266  ‘cfv 5270  (class class class)co 5943   ∈ cmpo 5945  Topctop 14440  TopOnctopon 14453   Cn ccn 14628   ×_t ctx 14695

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-map 6736  df-topgen 13063  df-top 14441  df-topon 14454  df-bases 14486  df-cn 14631  df-tx 14696

This theorem is referenced by:  cnmpt22  14737  txhmeo  14762  txswaphmeo  14764