Theorem tx2cn 23672

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 7997	. . 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 6693	. . . . . . . 8 ⊢ ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌 → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
4		elpreima 7041	. . . . . . . 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 6888	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd ‘𝑧))
7	6	eleq1d 2849	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ (2nd ‘𝑧) ∈ 𝑤))
8		1st2nd2 8011	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
9		xp1st 8004	. . . . . . . . . 10 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (1st ‘𝑧) ∈ 𝑋)
10		elxp6 8006	. . . . . . . . . . . 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 280	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑋 × 𝑤) ↔ ((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ 𝑋) ∧ (2nd ‘𝑧) ∈ 𝑤))
13	12	baib 543	. . . . . . . . . 10 ⊢ ((𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ (1st ‘𝑧) ∈ 𝑋) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (2nd ‘𝑧) ∈ 𝑤))
14	8, 9, 13	syl2anc 593	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (2nd ‘𝑧) ∈ 𝑤))
15	7, 14	bitr4d 284	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → (((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ 𝑧 ∈ (𝑋 × 𝑤)))
16	15	pm5.32i 582	. . . . . . 7 ⊢ ((𝑧 ∈ (𝑋 × 𝑌) ∧ ((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤)))
17	5, 16	bitri 277	. . . . . 6 ⊢ (𝑧 ∈ (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤)))
18		toponss 22989	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑤 ∈ 𝑆) → 𝑤 ⊆ 𝑌)
19	18	adantll 724	. . . . . . . . 9 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → 𝑤 ⊆ 𝑌)
20		xpss2 5669	. . . . . . . . 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 3937	. . . . . . 7 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ (𝑋 × 𝑤) → 𝑧 ∈ (𝑋 × 𝑌)))
23	22	pm4.71rd 570	. . . . . 6 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤))))
24	17, 23	bitr4id 292	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (𝑧 ∈ (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ 𝑧 ∈ (𝑋 × 𝑤)))
25	24	eqrdv 2762	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) = (𝑋 × 𝑤))
26		toponmax 22988	. . . . . 6 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝑅)
27		txopn 23664	. . . . . . 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 699	. . . . 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 2864	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤 ∈ 𝑆) → (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
32	31	ralrimiva 3156	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤 ∈ 𝑆 (◡(2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
33		txtopon 23653	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
34		iscn 23297	. . 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 597	. 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 723	1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ⊆ wss 3906  ⟨cop 4590   × cxp 5647  ^◡ccnv 5648   ↾ cres 5651   “ cima 5652   Fn wfn 6518  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398  1^st c1st 7970  2^nd c2nd 7971  TopOnctopon 22972   Cn ccn 23286   ×_t ctx 23622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-map 8812  df-topgen 17474  df-top 22956  df-topon 22973  df-bases 23008  df-cn 23289  df-tx 23624

This theorem is referenced by:  txcn  23688  txcmpb  23706  txkgen  23714  cnmpt2nd  23731  sxbrsiga  34589  txsconnlem  35595  txsconn  35596  hausgraph  43787