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Theorem tx1cn 22770

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 7844	. . 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 6592	. . . . . . . 8 ⊢ ((1^st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 → (1^st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
4		elpreima 6927	. . . . . . . 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 6785	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((1^st ↾ (𝑋 × 𝑌))‘𝑧) = (1^st ‘𝑧))
7	6	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ (1^st ‘𝑧) ∈ 𝑤))
8		1st2nd2 7859	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
9		xp2nd 7853	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑧) ∈ 𝑌)
10		elxp6 7854	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ ((1^st ‘𝑧) ∈ 𝑤 ∧ (2^nd ‘𝑧) ∈ 𝑌)))
11		anass 469	. . . . . . . . . . . 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 643	. . . . . . . . . . . 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 536	. . . . . . . . . 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 281	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ 𝑧 ∈ (𝑤 × 𝑌)))
17	16	pm5.32i 575	. . . . . . 7 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ ((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
18	5, 17	bitri 274	. . . . . 6 ⊢ (𝑧 ∈ (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
19		toponss 22086	. . . . . . . . . 10 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝑅) → 𝑤 ⊆ 𝑋)
20	19	adantlr 712	. . . . . . . . 9 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → 𝑤 ⊆ 𝑋)
21		xpss1 5603	. . . . . . . . 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 3919	. . . . . . 7 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑧 ∈ (𝑤 × 𝑌) → 𝑧 ∈ (𝑋 × 𝑌)))
24	23	pm4.71rd 563	. . . . . 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 2736	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) = (𝑤 × 𝑌))
27		toponmax 22085	. . . . . 6 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝑆)
28	27	ad2antlr 724	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → 𝑌 ∈ 𝑆)
29		txopn 22763	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑤 ∈ 𝑅 ∧ 𝑌 ∈ 𝑆)) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
30	29	anassrs 468	. . . . 5 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) ∧ 𝑌 ∈ 𝑆) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
31	28, 30	mpdan 684	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
32	26, 31	eqeltrd 2839	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×_t 𝑆))
33	32	ralrimiva 3108	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤 ∈ 𝑅 (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×_t 𝑆))
34		txtopon 22752	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
35		simpl 483	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ (TopOn‘𝑋))
36		iscn 22396	. . 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 710	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 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ⊆ wss 3886 ⟨cop 4567 × cxp 5582 ^◡ccnv 5583 ↾ cres 5586 “ cima 5587 Fn wfn 6421 ⟶wf 6422 ‘cfv 6426 (class class class)co 7267 1^st c1st 7818 2^nd c2nd 7819 TopOnctopon 22069 Cn ccn 22385 ×_t ctx 22721

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-fv 6434 df-ov 7270 df-oprab 7271 df-mpo 7272 df-1st 7820 df-2nd 7821 df-map 8604 df-topgen 17164 df-top 22053 df-topon 22070 df-bases 22106 df-cn 22388 df-tx 22723

This theorem is referenced by: txcn 22787 txcmpb 22805 cnmpt1st 22829 sxbrsiga 32265 txsconnlem 33210 txsconn 33211 hausgraph 41045

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