Theorem tx2cn 22507

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

Assertion

Ref	Expression
tx2cn	⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))

Proof of Theorem tx2cn

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f2ndres 7786	. . 3 ⊢ (2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌
2	1	a1i 11	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌)
3		ffn 6545	. . . . . . . 8 ⊢ ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌 → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
4		elpreima 6878	. . . . . . . 8 ⊢ ((2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) → (𝑧 ∈ (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤)))
5	1, 3, 4	mp2b 10	. . . . . . 7 ⊢ (𝑧 ∈ (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤))
6		fvres 6736	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd ‘𝑧))
7	6	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ (2nd ‘𝑧) ∈ 𝑤))
8		1st2nd2 7800	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
9		xp1st 7793	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (1st ‘𝑧) ∈ 𝑋)
10		elxp6 7795	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑤) ↔ (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ 𝑋 ∧ (2nd ‘𝑧) ∈ 𝑤)))
11		anass 472	. . . . . . . . . . . 12 ⊢ (((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ 𝑋) ∧ (2nd ‘𝑧) ∈ 𝑤) ↔ (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ 𝑋 ∧ (2nd ‘𝑧) ∈ 𝑤)))
12	10, 11	bitr4i 281	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑋 × 𝑤) ↔ ((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ 𝑋) ∧ (2nd ‘𝑧) ∈ 𝑤))
13	12	baib 539	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ 𝑋) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (2nd ‘𝑧) ∈ 𝑤))
14	8, 9, 13	syl2anc 587	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (2nd ‘𝑧) ∈ 𝑤))
15	7, 14	bitr4d 285	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ 𝑧 ∈ (𝑋 × 𝑤)))
16	15	pm5.32i 578	. . . . . . 7 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ ((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤)))
17	5, 16	bitri 278	. . . . . 6 ⊢ (𝑧 ∈ (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤)))
18		toponss 21824	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑤 ∈ 𝑆) → 𝑤 ⊆ 𝑌)
19	18	adantll 714	. . . . . . . . 9 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → 𝑤 ⊆ 𝑌)
20		xpss2 5571	. . . . . . . . 9 ⊢ (𝑤 ⊆ 𝑌 → (𝑋 × 𝑤) ⊆ (𝑋 × 𝑌))
21	19, 20	syl 17	. . . . . . . 8 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (𝑋 × 𝑤) ⊆ (𝑋 × 𝑌))
22	21	sseld 3900	. . . . . . 7 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ (𝑋 × 𝑤) → 𝑧 ∈ (𝑋 × 𝑌)))
23	22	pm4.71rd 566	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤))))
24	17, 23	bitr4id 293	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ 𝑧 ∈ (𝑋 × 𝑤)))
25	24	eqrdv 2735	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) = (𝑋 × 𝑤))
26		toponmax 21823	. . . . . 6 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝑅)
27		txopn 22499	. . . . . . 7 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑤 ∈ 𝑆)) → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆))
28	27	expr 460	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑋 ∈ 𝑅) → (𝑤 ∈ 𝑆 → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆)))
29	26, 28	mpidan 689	. . . . 5 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑤 ∈ 𝑆 → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆)))
30	29	imp 410	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆))
31	25, 30	eqeltrd 2838	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
32	31	ralrimiva 3105	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤 ∈ 𝑆 (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
33		txtopon 22488	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
34		iscn 22132	. . 3 ⊢ (((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆) ↔ ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑤 ∈ 𝑆 (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))))
35	33, 34	sylancom 591	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆) ↔ ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑤 ∈ 𝑆 (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))))
36	2, 32, 35	mpbir2and 713	1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∀wral 3061   ⊆ wss 3866  ⟨cop 4547   × cxp 5549  ^◡ccnv 5550   ↾ cres 5553   “ cima 5554   Fn wfn 6375  ⟶wf 6376  ‘cfv 6380  (class class class)co 7213  1^st c1st 7759  2^nd c2nd 7760  TopOnctopon 21807   Cn ccn 22121   ×_t ctx 22457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-map 8510  df-topgen 16948  df-top 21791  df-topon 21808  df-bases 21843  df-cn 22124  df-tx 22459

This theorem is referenced by:  txcn  22523  txcmpb  22541  txkgen  22549  cnmpt2nd  22566  sxbrsiga  31969  txsconnlem  32915  txsconn  32916  hausgraph  40740