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Theorem tx1cn 23633
Description: Continuity of the first projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
tx1cn ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))

Proof of Theorem tx1cn
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1stres 8037 . . 3 (1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋
21a1i 11 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋)
3 ffn 6737 . . . . . . . 8 ((1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
4 elpreima 7078 . . . . . . . 8 ((1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) → (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤)))
51, 3, 4mp2b 10 . . . . . . 7 (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤))
6 fvres 6926 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → ((1st ↾ (𝑋 × 𝑌))‘𝑧) = (1st𝑧))
76eleq1d 2824 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) → (((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ (1st𝑧) ∈ 𝑤))
8 1st2nd2 8052 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
9 xp2nd 8046 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (2nd𝑧) ∈ 𝑌)
10 elxp6 8047 . . . . . . . . . . . 12 (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑤 ∧ (2nd𝑧) ∈ 𝑌)))
11 anass 468 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑤) ∧ (2nd𝑧) ∈ 𝑌) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑤 ∧ (2nd𝑧) ∈ 𝑌)))
12 an32 646 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑤) ∧ (2nd𝑧) ∈ 𝑌) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑌) ∧ (1st𝑧) ∈ 𝑤))
1310, 11, 123bitr2i 299 . . . . . . . . . . 11 (𝑧 ∈ (𝑤 × 𝑌) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑌) ∧ (1st𝑧) ∈ 𝑤))
1413baib 535 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1st𝑧) ∈ 𝑤))
158, 9, 14syl2anc 584 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1st𝑧) ∈ 𝑤))
167, 15bitr4d 282 . . . . . . . 8 (𝑧 ∈ (𝑋 × 𝑌) → (((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤𝑧 ∈ (𝑤 × 𝑌)))
1716pm5.32i 574 . . . . . . 7 ((𝑧 ∈ (𝑋 × 𝑌) ∧ ((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
185, 17bitri 275 . . . . . 6 (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
19 toponss 22949 . . . . . . . . . 10 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑤𝑅) → 𝑤𝑋)
2019adantlr 715 . . . . . . . . 9 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → 𝑤𝑋)
21 xpss1 5708 . . . . . . . . 9 (𝑤𝑋 → (𝑤 × 𝑌) ⊆ (𝑋 × 𝑌))
2220, 21syl 17 . . . . . . . 8 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑤 × 𝑌) ⊆ (𝑋 × 𝑌))
2322sseld 3994 . . . . . . 7 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑧 ∈ (𝑤 × 𝑌) → 𝑧 ∈ (𝑋 × 𝑌)))
2423pm4.71rd 562 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌))))
2518, 24bitr4id 290 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ 𝑧 ∈ (𝑤 × 𝑌)))
2625eqrdv 2733 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) = (𝑤 × 𝑌))
27 toponmax 22948 . . . . . 6 (𝑆 ∈ (TopOn‘𝑌) → 𝑌𝑆)
2827ad2antlr 727 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → 𝑌𝑆)
29 txopn 23626 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑤𝑅𝑌𝑆)) → (𝑤 × 𝑌) ∈ (𝑅 ×t 𝑆))
3029anassrs 467 . . . . 5 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) ∧ 𝑌𝑆) → (𝑤 × 𝑌) ∈ (𝑅 ×t 𝑆))
3128, 30mpdan 687 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑤 × 𝑌) ∈ (𝑅 ×t 𝑆))
3226, 31eqeltrd 2839 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
3332ralrimiva 3144 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤𝑅 ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
34 txtopon 23615 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
35 simpl 482 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ (TopOn‘𝑋))
36 iscn 23259 . . 3 (((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑅 ∈ (TopOn‘𝑋)) → ((1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅) ↔ ((1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 ∧ ∀𝑤𝑅 ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))))
3734, 35, 36syl2anc 584 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅) ↔ ((1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 ∧ ∀𝑤𝑅 ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))))
382, 33, 37mpbir2and 713 1 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wss 3963  cop 4637   × cxp 5687  ccnv 5688  cres 5691  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  TopOnctopon 22932   Cn ccn 23248   ×t ctx 23584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cn 23251  df-tx 23586
This theorem is referenced by:  txcn  23650  txcmpb  23668  cnmpt1st  23692  sxbrsiga  34272  txsconnlem  35225  txsconn  35226  hausgraph  43194
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