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Theorem tx1cn 23524

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 7945	. . 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 6651	. . . . . . . 8 ⊢ ((1^st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 → (1^st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
4		elpreima 6991	. . . . . . . 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 6841	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((1^st ↾ (𝑋 × 𝑌))‘𝑧) = (1^st ‘𝑧))
7	6	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ (1^st ‘𝑧) ∈ 𝑤))
8		1st2nd2 7960	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
9		xp2nd 7954	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑧) ∈ 𝑌)
10		elxp6 7955	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ ((1^st ‘𝑧) ∈ 𝑤 ∧ (2^nd ‘𝑧) ∈ 𝑌)))
11		anass 468	. . . . . . . . . . . 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 646	. . . . . . . . . . . 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 299	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑤 × 𝑌) ↔ ((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (2^nd ‘𝑧) ∈ 𝑌) ∧ (1^st ‘𝑧) ∈ 𝑤))
14	13	baib 535	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (2^nd ‘𝑧) ∈ 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1^st ‘𝑧) ∈ 𝑤))
15	8, 9, 14	syl2anc 584	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1^st ‘𝑧) ∈ 𝑤))
16	7, 15	bitr4d 282	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ 𝑧 ∈ (𝑤 × 𝑌)))
17	16	pm5.32i 574	. . . . . . 7 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ ((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
18	5, 17	bitri 275	. . . . . 6 ⊢ (𝑧 ∈ (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
19		toponss 22842	. . . . . . . . . 10 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝑅) → 𝑤 ⊆ 𝑋)
20	19	adantlr 715	. . . . . . . . 9 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → 𝑤 ⊆ 𝑋)
21		xpss1 5633	. . . . . . . . 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 3928	. . . . . . 7 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑧 ∈ (𝑤 × 𝑌) → 𝑧 ∈ (𝑋 × 𝑌)))
24	23	pm4.71rd 562	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌))))
25	18, 24	bitr4id 290	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑧 ∈ (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ 𝑧 ∈ (𝑤 × 𝑌)))
26	25	eqrdv 2729	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) = (𝑤 × 𝑌))
27		toponmax 22841	. . . . . 6 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝑆)
28	27	ad2antlr 727	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → 𝑌 ∈ 𝑆)
29		txopn 23517	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑤 ∈ 𝑅 ∧ 𝑌 ∈ 𝑆)) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
30	29	anassrs 467	. . . . 5 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) ∧ 𝑌 ∈ 𝑆) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
31	28, 30	mpdan 687	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
32	26, 31	eqeltrd 2831	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×_t 𝑆))
33	32	ralrimiva 3124	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤 ∈ 𝑅 (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×_t 𝑆))
34		txtopon 23506	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
35		simpl 482	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ (TopOn‘𝑋))
36		iscn 23150	. . 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 584	. 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 713	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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ⊆ wss 3897 ⟨cop 4579 × cxp 5612 ^◡ccnv 5613 ↾ cres 5616 “ cima 5617 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 1^st c1st 7919 2^nd c2nd 7920 TopOnctopon 22825 Cn ccn 23139 ×_t ctx 23475

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-map 8752 df-topgen 17347 df-top 22809 df-topon 22826 df-bases 22861 df-cn 23142 df-tx 23477

This theorem is referenced by: txcn 23541 txcmpb 23559 cnmpt1st 23583 sxbrsiga 34303 txsconnlem 35284 txsconn 35285 hausgraph 43297

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