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Theorem txopn 21315
Description: The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
txopn (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆))

Proof of Theorem txopn
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . . . 6 ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
21txbasex 21279 . . . . 5 ((𝑅𝑉𝑆𝑊) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
3 bastg 20681 . . . . 5 (ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ V → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
42, 3syl 17 . . . 4 ((𝑅𝑉𝑆𝑊) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
54adantr 481 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
6 eqid 2621 . . . . . 6 (𝐴 × 𝐵) = (𝐴 × 𝐵)
7 xpeq1 5088 . . . . . . . 8 (𝑢 = 𝐴 → (𝑢 × 𝑣) = (𝐴 × 𝑣))
87eqeq2d 2631 . . . . . . 7 (𝑢 = 𝐴 → ((𝐴 × 𝐵) = (𝑢 × 𝑣) ↔ (𝐴 × 𝐵) = (𝐴 × 𝑣)))
9 xpeq2 5089 . . . . . . . 8 (𝑣 = 𝐵 → (𝐴 × 𝑣) = (𝐴 × 𝐵))
109eqeq2d 2631 . . . . . . 7 (𝑣 = 𝐵 → ((𝐴 × 𝐵) = (𝐴 × 𝑣) ↔ (𝐴 × 𝐵) = (𝐴 × 𝐵)))
118, 10rspc2ev 3308 . . . . . 6 ((𝐴𝑅𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐴 × 𝐵)) → ∃𝑢𝑅𝑣𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣))
126, 11mp3an3 1410 . . . . 5 ((𝐴𝑅𝐵𝑆) → ∃𝑢𝑅𝑣𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣))
13 xpexg 6913 . . . . . 6 ((𝐴𝑅𝐵𝑆) → (𝐴 × 𝐵) ∈ V)
14 eqid 2621 . . . . . . 7 (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
1514elrnmpt2g 6725 . . . . . 6 ((𝐴 × 𝐵) ∈ V → ((𝐴 × 𝐵) ∈ ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ↔ ∃𝑢𝑅𝑣𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣)))
1613, 15syl 17 . . . . 5 ((𝐴𝑅𝐵𝑆) → ((𝐴 × 𝐵) ∈ ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ↔ ∃𝑢𝑅𝑣𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣)))
1712, 16mpbird 247 . . . 4 ((𝐴𝑅𝐵𝑆) → (𝐴 × 𝐵) ∈ ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)))
1817adantl 482 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)))
195, 18sseldd 3584 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
201txval 21277 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
2120adantr 481 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
2219, 21eleqtrrd 2701 1 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  wss 3555   × cxp 5072  ran crn 5075  cfv 5847  (class class class)co 6604  cmpt2 6606  topGenctg 16019   ×t ctx 21273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-topgen 16025  df-tx 21275
This theorem is referenced by:  txcld  21316  txbasval  21319  neitx  21320  tx1cn  21322  tx2cn  21323  txlly  21349  txnlly  21350  txhaus  21360  txlm  21361  tx1stc  21363  txkgen  21365  xkococnlem  21372  cxpcn3  24389  cvmlift2lem11  31000  cvmlift2lem12  31001
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