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Theorem tx1cn 23607

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 8029	. . 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 6730	. . . . . . . 8 ⊢ ((1^st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 → (1^st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
4		elpreima 7073	. . . . . . . 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 6922	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((1^st ↾ (𝑋 × 𝑌))‘𝑧) = (1^st ‘𝑧))
7	6	eleq1d 2811	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ (1^st ‘𝑧) ∈ 𝑤))
8		1st2nd2 8044	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
9		xp2nd 8038	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑧) ∈ 𝑌)
10		elxp6 8039	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ ((1^st ‘𝑧) ∈ 𝑤 ∧ (2^nd ‘𝑧) ∈ 𝑌)))
11		anass 467	. . . . . . . . . . . 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 644	. . . . . . . . . . . 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 298	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑤 × 𝑌) ↔ ((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (2^nd ‘𝑧) ∈ 𝑌) ∧ (1^st ‘𝑧) ∈ 𝑤))
14	13	baib 534	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (2^nd ‘𝑧) ∈ 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1^st ‘𝑧) ∈ 𝑤))
15	8, 9, 14	syl2anc 582	. . . . . . . . 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 573	. . . . . . 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 22923	. . . . . . . . . 10 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝑅) → 𝑤 ⊆ 𝑋)
20	19	adantlr 713	. . . . . . . . 9 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → 𝑤 ⊆ 𝑋)
21		xpss1 5703	. . . . . . . . 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 3978	. . . . . . 7 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑧 ∈ (𝑤 × 𝑌) → 𝑧 ∈ (𝑋 × 𝑌)))
24	23	pm4.71rd 561	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌))))
25	18, 24	bitr4id 289	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑧 ∈ (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ 𝑧 ∈ (𝑤 × 𝑌)))
26	25	eqrdv 2724	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) = (𝑤 × 𝑌))
27		toponmax 22922	. . . . . 6 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝑆)
28	27	ad2antlr 725	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → 𝑌 ∈ 𝑆)
29		txopn 23600	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑤 ∈ 𝑅 ∧ 𝑌 ∈ 𝑆)) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
30	29	anassrs 466	. . . . 5 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) ∧ 𝑌 ∈ 𝑆) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
31	28, 30	mpdan 685	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
32	26, 31	eqeltrd 2826	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×_t 𝑆))
33	32	ralrimiva 3136	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤 ∈ 𝑅 (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×_t 𝑆))
34		txtopon 23589	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
35		simpl 481	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ (TopOn‘𝑋))
36		iscn 23233	. . 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 582	. 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 711	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 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ⊆ wss 3947 ⟨cop 4639 × cxp 5682 ^◡ccnv 5683 ↾ cres 5686 “ cima 5687 Fn wfn 6551 ⟶wf 6552 ‘cfv 6556 (class class class)co 7426 1^st c1st 8003 2^nd c2nd 8004 TopOnctopon 22906 Cn ccn 23222 ×_t ctx 23558

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-fv 6564 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8005 df-2nd 8006 df-map 8859 df-topgen 17460 df-top 22890 df-topon 22907 df-bases 22943 df-cn 23225 df-tx 23560

This theorem is referenced by: txcn 23624 txcmpb 23642 cnmpt1st 23666 sxbrsiga 34126 txsconnlem 35070 txsconn 35071 hausgraph 42888

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