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Theorem txopn 21603
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 2756 . . . . . 6 ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
21txbasex 21567 . . . . 5 ((𝑅𝑉𝑆𝑊) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
3 bastg 20968 . . . . 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 472 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
6 eqid 2756 . . . . . 6 (𝐴 × 𝐵) = (𝐴 × 𝐵)
7 xpeq1 5276 . . . . . . . 8 (𝑢 = 𝐴 → (𝑢 × 𝑣) = (𝐴 × 𝑣))
87eqeq2d 2766 . . . . . . 7 (𝑢 = 𝐴 → ((𝐴 × 𝐵) = (𝑢 × 𝑣) ↔ (𝐴 × 𝐵) = (𝐴 × 𝑣)))
9 xpeq2 5282 . . . . . . . 8 (𝑣 = 𝐵 → (𝐴 × 𝑣) = (𝐴 × 𝐵))
109eqeq2d 2766 . . . . . . 7 (𝑣 = 𝐵 → ((𝐴 × 𝐵) = (𝐴 × 𝑣) ↔ (𝐴 × 𝐵) = (𝐴 × 𝐵)))
118, 10rspc2ev 3459 . . . . . 6 ((𝐴𝑅𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐴 × 𝐵)) → ∃𝑢𝑅𝑣𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣))
126, 11mp3an3 1558 . . . . 5 ((𝐴𝑅𝐵𝑆) → ∃𝑢𝑅𝑣𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣))
13 xpexg 7121 . . . . . 6 ((𝐴𝑅𝐵𝑆) → (𝐴 × 𝐵) ∈ V)
14 eqid 2756 . . . . . . 7 (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
1514elrnmpt2g 6933 . . . . . 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 473 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)))
195, 18sseldd 3741 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
201txval 21565 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
2120adantr 472 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
2219, 21eleqtrrd 2838 1 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1628  wcel 2135  wrex 3047  Vcvv 3336  wss 3711   × cxp 5260  ran crn 5263  cfv 6045  (class class class)co 6809  cmpt2 6811  topGenctg 16296   ×t ctx 21561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-1st 7329  df-2nd 7330  df-topgen 16302  df-tx 21563
This theorem is referenced by:  txcld  21604  txbasval  21607  neitx  21608  tx1cn  21610  tx2cn  21611  txlly  21637  txnlly  21638  txhaus  21648  txlm  21649  tx1stc  21651  txkgen  21653  xkococnlem  21660  cxpcn3  24684  cvmlift2lem11  31598  cvmlift2lem12  31599
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