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Theorem elsx 30385
Description: The cartesian product of two open sets is an element of the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)
Assertion
Ref Expression
elsx (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ (𝑆 ×s 𝑇))

Proof of Theorem elsx
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . . . . . 6 ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
21txbasex 21417 . . . . 5 ((𝑆𝑉𝑇𝑊) → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
3 sssigagen 30336 . . . . 5 (ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ⊆ (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
42, 3syl 17 . . . 4 ((𝑆𝑉𝑇𝑊) → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ⊆ (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
54adantr 480 . . 3 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ⊆ (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
6 eqid 2651 . . . . . 6 (𝐴 × 𝐵) = (𝐴 × 𝐵)
7 xpeq1 5157 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
87eqeq2d 2661 . . . . . . 7 (𝑥 = 𝐴 → ((𝐴 × 𝐵) = (𝑥 × 𝑦) ↔ (𝐴 × 𝐵) = (𝐴 × 𝑦)))
9 xpeq2 5163 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
109eqeq2d 2661 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 × 𝐵) = (𝐴 × 𝑦) ↔ (𝐴 × 𝐵) = (𝐴 × 𝐵)))
118, 10rspc2ev 3355 . . . . . 6 ((𝐴𝑆𝐵𝑇 ∧ (𝐴 × 𝐵) = (𝐴 × 𝐵)) → ∃𝑥𝑆𝑦𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦))
126, 11mp3an3 1453 . . . . 5 ((𝐴𝑆𝐵𝑇) → ∃𝑥𝑆𝑦𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦))
13 xpexg 7002 . . . . . 6 ((𝐴𝑆𝐵𝑇) → (𝐴 × 𝐵) ∈ V)
14 eqid 2651 . . . . . . 7 (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
1514elrnmpt2g 6814 . . . . . 6 ((𝐴 × 𝐵) ∈ V → ((𝐴 × 𝐵) ∈ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ↔ ∃𝑥𝑆𝑦𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦)))
1613, 15syl 17 . . . . 5 ((𝐴𝑆𝐵𝑇) → ((𝐴 × 𝐵) ∈ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ↔ ∃𝑥𝑆𝑦𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦)))
1712, 16mpbird 247 . . . 4 ((𝐴𝑆𝐵𝑇) → (𝐴 × 𝐵) ∈ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
1817adantl 481 . . 3 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
195, 18sseldd 3637 . 2 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
201sxval 30381 . . 3 ((𝑆𝑉𝑇𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
2120adantr 480 . 2 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
2219, 21eleqtrrd 2733 1 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ (𝑆 ×s 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wrex 2942  Vcvv 3231  wss 3607   × cxp 5141  ran crn 5144  cfv 5926  (class class class)co 6690  cmpt2 6692  sigaGencsigagen 30329   ×s csx 30379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-siga 30299  df-sigagen 30330  df-sx 30380
This theorem is referenced by:  1stmbfm  30450  2ndmbfm  30451
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