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Theorem tx1cn 22216
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 7712 . . 3 (1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋
21a1i 11 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋)
3 toponss 21534 . . . . . . . . . 10 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑤𝑅) → 𝑤𝑋)
43adantlr 713 . . . . . . . . 9 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → 𝑤𝑋)
5 xpss1 5573 . . . . . . . . 9 (𝑤𝑋 → (𝑤 × 𝑌) ⊆ (𝑋 × 𝑌))
64, 5syl 17 . . . . . . . 8 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑤 × 𝑌) ⊆ (𝑋 × 𝑌))
76sseld 3965 . . . . . . 7 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑧 ∈ (𝑤 × 𝑌) → 𝑧 ∈ (𝑋 × 𝑌)))
87pm4.71rd 565 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌))))
9 ffn 6513 . . . . . . . 8 ((1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)⟶𝑋 → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
10 elpreima 6827 . . . . . . . 8 ((1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) → (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤)))
111, 9, 10mp2b 10 . . . . . . 7 (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ ((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤))
12 fvres 6688 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → ((1st ↾ (𝑋 × 𝑌))‘𝑧) = (1st𝑧))
1312eleq1d 2897 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) → (((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤 ↔ (1st𝑧) ∈ 𝑤))
14 1st2nd2 7727 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
15 xp2nd 7721 . . . . . . . . . 10 (𝑧 ∈ (𝑋 × 𝑌) → (2nd𝑧) ∈ 𝑌)
16 elxp6 7722 . . . . . . . . . . . 12 (𝑧 ∈ (𝑤 × 𝑌) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑤 ∧ (2nd𝑧) ∈ 𝑌)))
17 anass 471 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑤) ∧ (2nd𝑧) ∈ 𝑌) ↔ (𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ ((1st𝑧) ∈ 𝑤 ∧ (2nd𝑧) ∈ 𝑌)))
18 an32 644 . . . . . . . . . . . 12 (((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (1st𝑧) ∈ 𝑤) ∧ (2nd𝑧) ∈ 𝑌) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑌) ∧ (1st𝑧) ∈ 𝑤))
1916, 17, 183bitr2i 301 . . . . . . . . . . 11 (𝑧 ∈ (𝑤 × 𝑌) ↔ ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑌) ∧ (1st𝑧) ∈ 𝑤))
2019baib 538 . . . . . . . . . 10 ((𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩ ∧ (2nd𝑧) ∈ 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1st𝑧) ∈ 𝑤))
2114, 15, 20syl2anc 586 . . . . . . . . 9 (𝑧 ∈ (𝑋 × 𝑌) → (𝑧 ∈ (𝑤 × 𝑌) ↔ (1st𝑧) ∈ 𝑤))
2213, 21bitr4d 284 . . . . . . . 8 (𝑧 ∈ (𝑋 × 𝑌) → (((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤𝑧 ∈ (𝑤 × 𝑌)))
2322pm5.32i 577 . . . . . . 7 ((𝑧 ∈ (𝑋 × 𝑌) ∧ ((1st ↾ (𝑋 × 𝑌))‘𝑧) ∈ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
2411, 23bitri 277 . . . . . 6 (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ (𝑧 ∈ (𝑋 × 𝑌) ∧ 𝑧 ∈ (𝑤 × 𝑌)))
258, 24syl6rbbr 292 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑧 ∈ ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ↔ 𝑧 ∈ (𝑤 × 𝑌)))
2625eqrdv 2819 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) = (𝑤 × 𝑌))
27 toponmax 21533 . . . . . 6 (𝑆 ∈ (TopOn‘𝑌) → 𝑌𝑆)
2827ad2antlr 725 . . . . 5 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → 𝑌𝑆)
29 txopn 22209 . . . . . 6 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑤𝑅𝑌𝑆)) → (𝑤 × 𝑌) ∈ (𝑅 ×t 𝑆))
3029anassrs 470 . . . . 5 ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) ∧ 𝑌𝑆) → (𝑤 × 𝑌) ∈ (𝑅 ×t 𝑆))
3128, 30mpdan 685 . . . 4 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → (𝑤 × 𝑌) ∈ (𝑅 ×t 𝑆))
3226, 31eqeltrd 2913 . . 3 (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑤𝑅) → ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
3332ralrimiva 3182 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑤𝑅 ((1st ↾ (𝑋 × 𝑌)) “ 𝑤) ∈ (𝑅 ×t 𝑆))
34 txtopon 22198 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
35 simpl 485 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ (TopOn‘𝑋))
36 iscn 21842 . . 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 586 . 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 711 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 208  wa 398   = wceq 1533  wcel 2110  wral 3138  wss 3935  cop 4572   × cxp 5552  ccnv 5553  cres 5556  cima 5557   Fn wfn 6349  wf 6350  cfv 6354  (class class class)co 7155  1st c1st 7686  2nd c2nd 7687  TopOnctopon 21517   Cn ccn 21831   ×t ctx 22167
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-map 8407  df-topgen 16716  df-top 21501  df-topon 21518  df-bases 21553  df-cn 21834  df-tx 22169
This theorem is referenced by:  txcn  22233  txcmpb  22251  cnmpt1st  22275  sxbrsiga  31548  txsconnlem  32487  txsconn  32488  hausgraph  39812
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