Theorem cnmptcom 13092

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 13056	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
4	1, 2, 3	syl2anc 409	. . . . . . . 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 12996	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
7	5, 6	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top)
8		toptopon2 12811	. . . . . . . . 9 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
9	7, 8	sylib 121	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
10		cnf2 12999	. . . . . . . 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 1233	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶∪ 𝐿)
12		eqid 2170	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
13	12	fmpo 6180	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶∪ 𝐿)
14		ralcom 2633	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 ↔ ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
15	13, 14	bitr3i 185	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶∪ 𝐿 ↔ ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
16	11, 15	sylib 121	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
17		eqid 2170	. . . . . . 7 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
18	17	fmpo 6180	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 ↔ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
19	16, 18	sylib 121	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
20	19	ffnd 5348	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) Fn (𝑌 × 𝑋))
21		fnovim 5961	. . . 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	syl 14	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)))
23		nfcv 2312	. . . . . . 7 ⊢ Ⅎ𝑦𝑧
24		nfcv 2312	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
25		nfcv 2312	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
26		nfv 1521	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
27		nfcv 2312	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑥
28		nfmpo2 5921	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
29	27, 28, 23	nfov 5883	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧)
30		nfmpo1 5920	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
31	23, 30, 27	nfov 5883	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)
32	29, 31	nfeq 2320	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)
33	26, 32	nfim 1565	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
34		nfv 1521	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
35		nfmpo1 5920	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
36	25, 35, 24	nfov 5883	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧)
37		nfmpo2 5921	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
38	24, 37, 25	nfov 5883	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
39	36, 38	nfeq 2320	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
40	34, 39	nfim 1565	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 → (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
41		oveq2 5861	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧))
42		oveq1 5860	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
43	41, 42	eqeq12d 2185	. . . . . . . 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 229	. . . . . . 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 5860	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧))
46		oveq2 5861	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
47	45, 46	eqeq12d 2185	. . . . . . . 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 229	. . . . . . 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 2520	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 → ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐿))
50	49, 16	syl11 31	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → (𝜑 → 𝐴 ∈ ∪ 𝐿))
51	12	ovmpt4g 5975	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
52	51	3com12 1202	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
53	17	ovmpt4g 5975	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = 𝐴)
54	52, 53	eqtr4d 2206	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
55	54	3expia 1200	. . . . . . . 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 45	. . . . . . 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 2797	. . . . . 6 ⊢ ((𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑋) → (𝜑 → (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)))
58	57	com12 30	. . . . 5 ⊢ (𝜑 → ((𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑋) → (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)))
59	58	3impib 1196	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑋) → (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
60	59	mpoeq3dva 5917	. . 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 2206	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧)))
62	2, 1	cnmpt2nd 13083	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ 𝑤) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
63	2, 1	cnmpt1st 13082	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ 𝑧) ∈ ((𝐾 ×t 𝐽) Cn 𝐾))
64	2, 1, 62, 63, 5	cnmpt22f 13089	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
65	61, 64	eqeltrd 2247	1 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 973   = wceq 1348   ∈ wcel 2141  ∀wral 2448  ∪ cuni 3796   × cxp 4609   Fn wfn 5193  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853   ∈ cmpo 5855  Topctop 12789  TopOnctopon 12802   Cn ccn 12979   ×_t ctx 13046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-topgen 12600  df-top 12790  df-topon 12803  df-bases 12835  df-cn 12982  df-tx 13047

This theorem is referenced by: (None)