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Theorem cnmptcom 22288

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 22201	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
4	1, 2, 3	syl2anc 586	. . . . . . . 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 21851	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐽 ×_t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
7	5, 6	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Top)
8		toptopon2 21528	. . . . . . . . 9 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
9	7, 8	sylib 220	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘∪ 𝐿))
10		cnf2 21859	. . . . . . . 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 1367	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶∪ 𝐿)
12		eqid 2823	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
13	12	fmpo 7768	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶∪ 𝐿)
14		ralcom 3356	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ ∪ 𝐿 ↔ ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
15	13, 14	bitr3i 279	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶∪ 𝐿 ↔ ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
16	11, 15	sylib 220	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿)
17		eqid 2823	. . . . . . 7 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
18	17	fmpo 7768	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 ↔ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
19	16, 18	sylib 220	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴):(𝑌 × 𝑋)⟶∪ 𝐿)
20	19	ffnd 6517	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) Fn (𝑌 × 𝑋))
21		fnov 7284	. . . 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 220	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)))
23		nfcv 2979	. . . . . . 7 ⊢ Ⅎ𝑦𝑧
24		nfcv 2979	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
25		nfcv 2979	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
26		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
27		nfcv 2979	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝑥
28		nfmpo2 7237	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
29	27, 28, 23	nfov 7188	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧)
30		nfmpo1 7236	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
31	23, 30, 27	nfov 7188	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)
32	29, 31	nfeq 2993	. . . . . . . 8 ⊢ Ⅎ𝑦(𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥)
33	26, 32	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
34		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
35		nfmpo1 7236	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
36	25, 35, 24	nfov 7188	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧)
37		nfmpo2 7237	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)
38	24, 37, 25	nfov 7188	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
39	36, 38	nfeq 2993	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)
40	34, 39	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑥(𝜑 → (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
41		oveq2 7166	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧))
42		oveq1 7165	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
43	41, 42	eqeq12d 2839	. . . . . . . 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 7165	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧))
46		oveq2 7166	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
47	45, 46	eqeq12d 2839	. . . . . . . 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 3215	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ 𝐿 → ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ 𝐿))
50	49, 16	syl11 33	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → (𝜑 → 𝐴 ∈ ∪ 𝐿))
51	12	ovmpt4g 7299	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
52	51	3com12 1119	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = 𝐴)
53	17	ovmpt4g 7299	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥) = 𝐴)
54	52, 53	eqtr4d 2861	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ ∪ 𝐿) → (𝑥(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑦) = (𝑦(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑥))
55	54	3expia 1117	. . . . . . . 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 47	. . . . . . 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 3578	. . . . . 6 ⊢ ((𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑋) → (𝜑 → (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)))
58	57	com12 32	. . . . 5 ⊢ (𝜑 → ((𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑋) → (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤)))
59	58	3impib 1112	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑋) → (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧) = (𝑧(𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴)𝑤))
60	59	mpoeq3dva 7233	. . 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 2861	. 2 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧)))
62	2, 1	cnmpt2nd 22279	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ 𝑤) ∈ ((𝐾 ×_t 𝐽) Cn 𝐽))
63	2, 1	cnmpt1st 22278	. . 3 ⊢ (𝜑 → (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ 𝑧) ∈ ((𝐾 ×_t 𝐽) Cn 𝐾))
64	2, 1, 62, 63, 5	cnmpt22f 22285	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝑌, 𝑤 ∈ 𝑋 ↦ (𝑤(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)𝑧)) ∈ ((𝐾 ×_t 𝐽) Cn 𝐿))
65	61, 64	eqeltrd 2915	1 ⊢ (𝜑 → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐾 ×_t 𝐽) Cn 𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∪ cuni 4840 × cxp 5555 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 Topctop 21503 TopOnctopon 21520 Cn ccn 21834 ×_t ctx 22170

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-topgen 16719 df-top 21504 df-topon 21521 df-bases 21556 df-cn 21837 df-tx 22172

This theorem is referenced by: cnmpt2k 22298 htpycc 23586

W3C validator