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Theorem tx2cn 23618
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 8039 . . 3 (2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌
21a1i 11 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌)
3 ffn 6736 . . . . . . . 8 ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑌 → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
4 elpreima 7078 . . . . . . . 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 6925 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd𝑧))
76eleq1d 2826 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) → (((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ (2nd𝑧) ∈ 𝑤))
8 1st2nd2 8053 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
9 xp1st 8046 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (1st𝑧) ∈ 𝑋)
10 elxp6 8048 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑤) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑋 ∧ (2nd𝑧) ∈ 𝑤)))
11 anass 468 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑋) ∧ (2nd𝑧) ∈ 𝑤) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑋 ∧ (2nd𝑧) ∈ 𝑤)))
1210, 11bitr4i 278 . . . . . . . . . . 11 (𝑧 ∈ (𝑋 × 𝑤) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑋) ∧ (2nd𝑧) ∈ 𝑤))
1312baib 535 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑋) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (2nd𝑧) ∈ 𝑤))
148, 9, 13syl2anc 584 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (2nd𝑧) ∈ 𝑤))
157, 14bitr4d 282 . . . . . . . 8 (𝑧 ∈ (𝑋 × 𝑌) → (((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤𝑧 ∈ (𝑋 × 𝑤)))
1615pm5.32i 574 . . . . . . 7 ((𝑧 ∈ (𝑋 × 𝑌) ∧ ((2nd ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤)))
175, 16bitri 275 . . . . . 6 (𝑧 ∈ ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤)))
18 toponss 22933 . . . . . . . . . 10 ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑤𝑆) → 𝑤𝑌)
1918adantll 714 . . . . . . . . 9 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → 𝑤𝑌)
20 xpss2 5705 . . . . . . . . 9 (𝑤𝑌 → (𝑋 × 𝑤) ⊆ (𝑋 × 𝑌))
2119, 20syl 17 . . . . . . . 8 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑋 × 𝑤) ⊆ (𝑋 × 𝑌))
2221sseld 3982 . . . . . . 7 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑧 ∈ (𝑋 × 𝑤) → 𝑧 ∈ (𝑋 × 𝑌)))
2322pm4.71rd 562 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑧 ∈ (𝑋 × 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑋 × 𝑤))))
2417, 23bitr4id 290 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑧 ∈ ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ 𝑧 ∈ (𝑋 × 𝑤)))
2524eqrdv 2735 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) = (𝑋 × 𝑤))
26 toponmax 22932 . . . . . 6 (𝑅 ∈ (TopOn‘𝑋) → 𝑋𝑅)
27 txopn 23610 . . . . . . 7 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑋𝑅𝑤𝑆)) → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆))
2827expr 456 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑋𝑅) → (𝑤𝑆 → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆)))
2926, 28mpidan 689 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑤𝑆 → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆)))
3029imp 406 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → (𝑋 × 𝑤) ∈ (𝑅 ×t 𝑆))
3125, 30eqeltrd 2841 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑆) → ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
3231ralrimiva 3146 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤𝑆 ((2nd ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
33 txtopon 23599 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
34 iscn 23243 . . 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 588 . 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 713 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 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wss 3951  cop 4632   × cxp 5683  ccnv 5684  cres 5687  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  TopOnctopon 22916   Cn ccn 23232   ×t ctx 23568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cn 23235  df-tx 23570
This theorem is referenced by:  txcn  23634  txcmpb  23652  txkgen  23660  cnmpt2nd  23677  sxbrsiga  34292  txsconnlem  35245  txsconn  35246  hausgraph  43217
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