tx1cn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tx1cn		Structured version Visualization version GIF version

Theorem tx1cn 22214

Description: Continuity of the first projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)

**Assertion**
Ref	Expression
tx1cn	⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1^st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×_t 𝑆) Cn 𝑅))

Proof of Theorem tx1cn Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1stres 7695	. . 3 ⊢ (1^st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋
2	1	a1i 11	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1^st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋)
3		ffn 6487	. . . . . . . 8 ⊢ ((1^st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 → (1^st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
4		elpreima 6805	. . . . . . . 8 ⊢ ((1^st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) → (𝑧 ∈ (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤)))
5	1, 3, 4	mp2b 10	. . . . . . 7 ⊢ (𝑧 ∈ (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤))
6		fvres 6664	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((1^st ↾ (𝑋 × 𝑌))‘𝑧) = (1^st ‘𝑧))
7	6	eleq1d 2874	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ (1^st ‘𝑧) ∈ 𝑤))
8		1st2nd2 7710	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
9		xp2nd 7704	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑧) ∈ 𝑌)
10		elxp6 7705	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ ((1^st ‘𝑧) ∈ 𝑤 ∧ (2^nd ‘𝑧) ∈ 𝑌)))
11		anass 472	. . . . . . . . . . . 12 ⊢ (((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (1^st ‘𝑧) ∈ 𝑤) ∧ (2^nd ‘𝑧) ∈ 𝑌) ↔ (𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ ((1^st ‘𝑧) ∈ 𝑤 ∧ (2^nd ‘𝑧) ∈ 𝑌)))
12		an32 645	. . . . . . . . . . . 12 ⊢ (((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (1^st ‘𝑧) ∈ 𝑤) ∧ (2^nd ‘𝑧) ∈ 𝑌) ↔ ((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (2^nd ‘𝑧) ∈ 𝑌) ∧ (1^st ‘𝑧) ∈ 𝑤))
13	10, 11, 12	3bitr2i 302	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑤 × 𝑌) ↔ ((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (2^nd ‘𝑧) ∈ 𝑌) ∧ (1^st ‘𝑧) ∈ 𝑤))
14	13	baib 539	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (2^nd ‘𝑧) ∈ 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1^st ‘𝑧) ∈ 𝑤))
15	8, 9, 14	syl2anc 587	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1^st ‘𝑧) ∈ 𝑤))
16	7, 15	bitr4d 285	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ 𝑧 ∈ (𝑤 × 𝑌)))
17	16	pm5.32i 578	. . . . . . 7 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ ((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
18	5, 17	bitri 278	. . . . . 6 ⊢ (𝑧 ∈ (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
19		toponss 21532	. . . . . . . . . 10 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝑅) → 𝑤 ⊆ 𝑋)
20	19	adantlr 714	. . . . . . . . 9 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → 𝑤 ⊆ 𝑋)
21		xpss1 5538	. . . . . . . . 9 ⊢ (𝑤 ⊆ 𝑋 → (𝑤 × 𝑌) ⊆ (𝑋 × 𝑌))
22	20, 21	syl 17	. . . . . . . 8 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑤 × 𝑌) ⊆ (𝑋 × 𝑌))
23	22	sseld 3914	. . . . . . 7 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑧 ∈ (𝑤 × 𝑌) → 𝑧 ∈ (𝑋 × 𝑌)))
24	23	pm4.71rd 566	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌))))
25	18, 24	bitr4id 293	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑧 ∈ (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ 𝑧 ∈ (𝑤 × 𝑌)))
26	25	eqrdv 2796	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) = (𝑤 × 𝑌))
27		toponmax 21531	. . . . . 6 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝑆)
28	27	ad2antlr 726	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → 𝑌 ∈ 𝑆)
29		txopn 22207	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑤 ∈ 𝑅 ∧ 𝑌 ∈ 𝑆)) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
30	29	anassrs 471	. . . . 5 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) ∧ 𝑌 ∈ 𝑆) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
31	28, 30	mpdan 686	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
32	26, 31	eqeltrd 2890	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×_t 𝑆))
33	32	ralrimiva 3149	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤 ∈ 𝑅 (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×_t 𝑆))
34		txtopon 22196	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
35		simpl 486	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ (TopOn‘𝑋))
36		iscn 21840	. . 3 ⊢ (((𝑅 ×_t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑅 ∈ (TopOn‘𝑋)) → ((1^st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×_t 𝑆) Cn 𝑅) ↔ ((1^st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 ∧ ∀𝑤 ∈ 𝑅 (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×_t 𝑆))))
37	34, 35, 36	syl2anc 587	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((1^st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×_t 𝑆) Cn 𝑅) ↔ ((1^st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 ∧ ∀𝑤 ∈ 𝑅 (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×_t 𝑆))))
38	2, 33, 37	mpbir2and 712	1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1^st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×_t 𝑆) Cn 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 ⟨cop 4531 × cxp 5517 ^◡ccnv 5518 ↾ cres 5521 “ cima 5522 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 1^st c1st 7669 2^nd c2nd 7670 TopOnctopon 21515 Cn ccn 21829 ×_t ctx 22165

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-topgen 16709 df-top 21499 df-topon 21516 df-bases 21551 df-cn 21832 df-tx 22167

This theorem is referenced by: txcn 22231 txcmpb 22249 cnmpt1st 22273 sxbrsiga 31658 txsconnlem 32600 txsconn 32601 hausgraph 40156

W3C validator