Theorem tx2cn 23591

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 7964	. . 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 6666	. . . . . . . 8 ⊢ ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌 → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
4		elpreima 7008	. . . . . . . 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 6857	. . . . . . . . . 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 7978	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
9		xp1st 7971	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (1st ‘𝑧) ∈ 𝑋)
10		elxp6 7973	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑤) ↔ (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ 𝑋 ∧ (2nd ‘𝑧) ∈ 𝑤)))
11		anass 468	. . . . . . . . . . . 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 278	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑋 × 𝑤) ↔ ((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ 𝑋) ∧ (2nd ‘𝑧) ∈ 𝑤))
13	12	baib 535	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ 𝑋) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (2nd ‘𝑧) ∈ 𝑤))
14	8, 9, 13	syl2anc 585	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (2nd ‘𝑧) ∈ 𝑤))
15	7, 14	bitr4d 282	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ 𝑧 ∈ (𝑋 × 𝑤)))
16	15	pm5.32i 574	. . . . . . 7 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ ((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤)))
17	5, 16	bitri 275	. . . . . 6 ⊢ (𝑧 ∈ (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤)))
18		toponss 22908	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑤 ∈ 𝑆) → 𝑤 ⊆ 𝑌)
19	18	adantll 715	. . . . . . . . 9 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → 𝑤 ⊆ 𝑌)
20		xpss2 5648	. . . . . . . . 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 3921	. . . . . . 7 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ (𝑋 × 𝑤) → 𝑧 ∈ (𝑋 × 𝑌)))
23	22	pm4.71rd 562	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤))))
24	17, 23	bitr4id 290	. . . . 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 22907	. . . . . 6 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝑅)
27		txopn 23583	. . . . . . 7 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑋 ∈ 𝑅 ∧ 𝑤 ∈ 𝑆)) → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆))
28	27	expr 456	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑋 ∈ 𝑅) → (𝑤 ∈ 𝑆 → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆)))
29	26, 28	mpidan 690	. . . . 5 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑤 ∈ 𝑆 → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆)))
30	29	imp 406	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆))
31	25, 30	eqeltrd 2837	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
32	31	ralrimiva 3130	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤 ∈ 𝑆 (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
33		txtopon 23572	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
34		iscn 23216	. . 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 589	. 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 714	1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  ⟨cop 4574   × cxp 5626  ^◡ccnv 5627   ↾ cres 5630   “ cima 5631   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7364  1^st c1st 7937  2^nd c2nd 7938  TopOnctopon 22891   Cn ccn 23205   ×_t ctx 23541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5306  ax-pr 5374  ax-un 7686

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5523  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7367  df-oprab 7368  df-mpo 7369  df-1st 7939  df-2nd 7940  df-map 8772  df-topgen 17403  df-top 22875  df-topon 22892  df-bases 22927  df-cn 23208  df-tx 23543

This theorem is referenced by:  txcn  23607  txcmpb  23625  txkgen  23633  cnmpt2nd  23650  sxbrsiga  34456  txsconnlem  35444  txsconn  35445  hausgraph  43659