Theorem cnmptcom 24052

Description: The argument converse of a continuous function is continuous. (Contributed by Mario Carneiro, 6-Jun-2014.)

Hypotheses

Ref	Expression
cnmptcom.3	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmptcom.4	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmptcom.6	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))

Assertion

Ref	Expression
cnmptcom	⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))

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

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

Proof of Theorem cnmptcom

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

Step	Hyp	Ref	Expression
1		cnmptcom.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		cnmptcom.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
3		txtopon 23965	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
4	1, 2, 3	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5		cnmptcom.6	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
6		cntop2 23614	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
7	5, 6	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top)
8		toptopon2 23291	. . . . . . . . 9 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
9	7, 8	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
10		cnf2 23622	. . . . . . . 8 ⊢ (((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)
11	4, 9, 5, 10	syl3anc 1399	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶∪ 𝐿)
12		eqid 2763	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
13	12	fmpo 8114	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶∪ 𝐿)
14		ralcom 3294	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 ↔ ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
15	13, 14	bitr3i 280	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶∪ 𝐿 ↔ ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
16	11, 15	sylib 221	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
17		eqid 2763	. . . . . . 7 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
18	17	fmpo 8114	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 ↔ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
19	16, 18	sylib 221	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
20	19	ffnd 6740	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) Fn (𝑌 × 𝑋))
21		fnov 7582	. . . 4 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) Fn (𝑌 × 𝑋) ↔ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)))
22	20, 21	sylib 221	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)))
23		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑦𝑧
24		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
25		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
26		nfv 1949	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
27		nfcv 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑥
28		nfmpo2 7532	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
29	27, 28, 23	nfov 7481	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧)
30		nfmpo1 7531	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
31	23, 30, 27	nfov 7481	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)
32	29, 31	nfeq 2939	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)
33	26, 32	nfim 1931	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
34		nfv 1949	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
35		nfmpo1 7531	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
36	25, 35, 24	nfov 7481	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧)
37		nfmpo2 7532	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
38	24, 37, 25	nfov 7481	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
39	36, 38	nfeq 2939	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
40	34, 39	nfim 1931	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 → (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
41		oveq2 7459	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧))
42		oveq1 7458	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
43	41, 42	eqeq12d 2779	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ((𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) ↔ (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)))
44	43	imbi2d 343	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝜑 → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)) ↔ (𝜑 → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))))
45		oveq1 7458	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧))
46		oveq2 7459	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
47	45, 46	eqeq12d 2779	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) ↔ (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)))
48	47	imbi2d 343	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝜑 → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)) ↔ (𝜑 → (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))))
49		rsp2 3283	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 → ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐿))
50	49, 16	syl11 34	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → (𝜑 → 𝐴 ∈ ∪ 𝐿))
51	12	ovmpt4g 7599	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
52	51	3com12 1141	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
53	17	ovmpt4g 7599	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = 𝐴)
54	52, 53	eqtr4d 2801	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
55	54	3expia 1139	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → (𝐴 ∈ ∪ 𝐿 → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)))
56	50, 55	syld 48	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → (𝜑 → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)))
57	23, 24, 25, 33, 40, 44, 48, 56	vtocl2gaf 3542	. . . . . 6 ⊢ ((𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑋) → (𝜑 → (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)))
58	57	com12 33	. . . . 5 ⊢ (𝜑 → ((𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑋) → (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)))
59	58	3impib 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑋) → (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
60	59	mpoeq3dva 7528	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧)) = (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)))
61	22, 60	eqtr4d 2801	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧)))
62	2, 1	cnmpt2nd 24043	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ 𝑤) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
63	2, 1	cnmpt1st 24042	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ 𝑧) ∈ ((𝐾 ×t 𝐽) Cn 𝐾))
64	2, 1, 62, 63, 5	cnmpt22f 24049	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
65	61, 64	eqeltrd 2864	1 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∪ cuni 4872   × cxp 5656   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  Topctop 23266  TopOnctopon 23283   Cn ccn 23597   ×_t ctx 23934

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fo 6572  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-map 8887  df-topgen 17664  df-top 23267  df-topon 23284  df-bases 23319  df-cn 23600  df-tx 23936

This theorem is used by:  cnmpt2k  24062  htpycc  25356