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Theorem tx2cn 22256
 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.
StepHypRef Expression
1 f2ndres 7709 . . 3 (2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌
21a1i 11 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌)
3 ffn 6495 . . . . . . . 8 ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌 → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
4 elpreima 6815 . . . . . . . 8 ((2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) → (𝑧 ∈ ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤)))
51, 3, 4mp2b 10 . . . . . . 7 (𝑧 ∈ ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤))
6 fvres 6674 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd𝑧))
76eleq1d 2874 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) → (((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ (2nd𝑧) ∈ 𝑤))
8 1st2nd2 7723 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
9 xp1st 7716 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (1st𝑧) ∈ 𝑋)
10 elxp6 7718 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑤) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑋 ∧ (2nd𝑧) ∈ 𝑤)))
11 anass 472 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑋) ∧ (2nd𝑧) ∈ 𝑤) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑋 ∧ (2nd𝑧) ∈ 𝑤)))
1210, 11bitr4i 281 . . . . . . . . . . 11 (𝑧 ∈ (𝑋 × 𝑤) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑋) ∧ (2nd𝑧) ∈ 𝑤))
1312baib 539 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑋) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (2nd𝑧) ∈ 𝑤))
148, 9, 13syl2anc 587 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (2nd𝑧) ∈ 𝑤))
157, 14bitr4d 285 . . . . . . . 8 (𝑧 ∈ (𝑋 × 𝑌) → (((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤𝑧 ∈ (𝑋 × 𝑤)))
1615pm5.32i 578 . . . . . . 7 ((𝑧 ∈ (𝑋 × 𝑌) ∧ ((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤)))
175, 16bitri 278 . . . . . 6 (𝑧 ∈ ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤)))
18 toponss 21573 . . . . . . . . . 10 ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑤𝑆) → 𝑤𝑌)
1918adantll 713 . . . . . . . . 9 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → 𝑤𝑌)
20 xpss2 5543 . . . . . . . . 9 (𝑤𝑌 → (𝑋 × 𝑤) ⊆ (𝑋 × 𝑌))
2119, 20syl 17 . . . . . . . 8 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑋 × 𝑤) ⊆ (𝑋 × 𝑌))
2221sseld 3916 . . . . . . 7 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑧 ∈ (𝑋 × 𝑤) → 𝑧 ∈ (𝑋 × 𝑌)))
2322pm4.71rd 566 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤))))
2417, 23bitr4id 293 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑧 ∈ ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ 𝑧 ∈ (𝑋 × 𝑤)))
2524eqrdv 2796 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) = (𝑋 × 𝑤))
26 toponmax 21572 . . . . . 6 (𝑅 ∈ (TopOn‘𝑋) → 𝑋𝑅)
27 txopn 22248 . . . . . . 7 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑋𝑅𝑤𝑆)) → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆))
2827expr 460 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑋𝑅) → (𝑤𝑆 → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆)))
2926, 28mpidan 688 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑤𝑆 → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆)))
3029imp 410 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆))
3125, 30eqeltrd 2890 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
3231ralrimiva 3149 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤𝑆 ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
33 txtopon 22237 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
34 iscn 21881 . . 3 (((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆) ↔ ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑤𝑆 ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))))
3533, 34sylancom 591 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆) ↔ ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑤𝑆 ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))))
362, 32, 35mpbir2and 712 1 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3883  ⟨cop 4534   × cxp 5521  ◡ccnv 5522   ↾ cres 5525   “ cima 5526   Fn wfn 6327  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145  1st c1st 7682  2nd c2nd 7683  TopOnctopon 21556   Cn ccn 21870   ×t ctx 22206 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7684  df-2nd 7685  df-map 8409  df-topgen 16729  df-top 21540  df-topon 21557  df-bases 21592  df-cn 21873  df-tx 22208 This theorem is referenced by:  txcn  22272  txcmpb  22290  txkgen  22298  cnmpt2nd  22315  sxbrsiga  31724  txsconnlem  32666  txsconn  32667  hausgraph  40327
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