tx1cn - Intuitionistic Logic Explorer

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Theorem tx1cn 14992

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 6321	. . 3 ⊢ (1^st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋
2	1	a1i 9	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1^st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋)
3		ffn 5482	. . . . . . . 8 ⊢ ((1^st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 → (1^st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
4		elpreima 5766	. . . . . . . 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 8	. . . . . . 7 ⊢ (𝑧 ∈ (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤))
6		fvres 5663	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((1^st ↾ (𝑋 × 𝑌))‘𝑧) = (1^st ‘𝑧))
7	6	eleq1d 2300	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ (1^st ‘𝑧) ∈ 𝑤))
8		1st2nd2 6337	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
9		xp2nd 6328	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (2^nd ‘𝑧) ∈ 𝑌)
10		elxp6 6331	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ ((1^st ‘𝑧) ∈ 𝑤 ∧ (2^nd ‘𝑧) ∈ 𝑌)))
11		anass 401	. . . . . . . . . . . 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 564	. . . . . . . . . . . 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 208	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑤 × 𝑌) ↔ ((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (2^nd ‘𝑧) ∈ 𝑌) ∧ (1^st ‘𝑧) ∈ 𝑤))
14	13	baib 926	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∧ (2^nd ‘𝑧) ∈ 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1^st ‘𝑧) ∈ 𝑤))
15	8, 9, 14	syl2anc 411	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1^st ‘𝑧) ∈ 𝑤))
16	7, 15	bitr4d 191	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ 𝑧 ∈ (𝑤 × 𝑌)))
17	16	pm5.32i 454	. . . . . . 7 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ ((1^st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
18	5, 17	bitri 184	. . . . . 6 ⊢ (𝑧 ∈ (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
19		toponss 14749	. . . . . . . . . 10 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝑅) → 𝑤 ⊆ 𝑋)
20	19	adantlr 477	. . . . . . . . 9 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → 𝑤 ⊆ 𝑋)
21		xpss1 4836	. . . . . . . . 9 ⊢ (𝑤 ⊆ 𝑋 → (𝑤 × 𝑌) ⊆ (𝑋 × 𝑌))
22	20, 21	syl 14	. . . . . . . 8 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑤 × 𝑌) ⊆ (𝑋 × 𝑌))
23	22	sseld 3226	. . . . . . 7 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑧 ∈ (𝑤 × 𝑌) → 𝑧 ∈ (𝑋 × 𝑌)))
24	23	pm4.71rd 394	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌))))
25	18, 24	bitr4id 199	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑧 ∈ (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ 𝑧 ∈ (𝑤 × 𝑌)))
26	25	eqrdv 2229	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) = (𝑤 × 𝑌))
27		toponmax 14748	. . . . . 6 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝑆)
28	27	ad2antlr 489	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → 𝑌 ∈ 𝑆)
29		txopn 14988	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑤 ∈ 𝑅 ∧ 𝑌 ∈ 𝑆)) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
30	29	anassrs 400	. . . . 5 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) ∧ 𝑌 ∈ 𝑆) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
31	28, 30	mpdan 421	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (𝑤 × 𝑌) ∈ (𝑅 ×_t 𝑆))
32	26, 31	eqeltrd 2308	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑅) → (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×_t 𝑆))
33	32	ralrimiva 2605	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤 ∈ 𝑅 (^◡(1^st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×_t 𝑆))
34		txtopon 14985	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×_t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
35		simpl 109	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ (TopOn‘𝑋))
36		iscn 14920	. . 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 411	. 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 952	1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1^st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×_t 𝑆) Cn 𝑅))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1397 ∈ wcel 2202 ∀wral 2510 ⊆ wss 3200 ⟨cop 3672 × cxp 4723 ^◡ccnv 4724 ↾ cres 4727 “ cima 4728 Fn wfn 5321 ⟶wf 5322 ‘cfv 5326 (class class class)co 6017 1^st c1st 6300 2^nd c2nd 6301 TopOnctopon 14733 Cn ccn 14908 ×_t ctx 14975

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-map 6818 df-topgen 13342 df-top 14721 df-topon 14734 df-bases 14766 df-cn 14911 df-tx 14976

This theorem is referenced by: txcn 14998 cnmpt1st 15011

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