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Theorem elsx 34175
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 2735 . . . . . 6 ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
21txbasex 23590 . . . . 5 ((𝑆𝑉𝑇𝑊) → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
3 sssigagen 34126 . . . . 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 2735 . . . . . 6 (𝐴 × 𝐵) = (𝐴 × 𝐵)
7 xpeq1 5703 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
87eqeq2d 2746 . . . . . . 7 (𝑥 = 𝐴 → ((𝐴 × 𝐵) = (𝑥 × 𝑦) ↔ (𝐴 × 𝐵) = (𝐴 × 𝑦)))
9 xpeq2 5710 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
109eqeq2d 2746 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 × 𝐵) = (𝐴 × 𝑦) ↔ (𝐴 × 𝐵) = (𝐴 × 𝐵)))
118, 10rspc2ev 3635 . . . . . 6 ((𝐴𝑆𝐵𝑇 ∧ (𝐴 × 𝐵) = (𝐴 × 𝐵)) → ∃𝑥𝑆𝑦𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦))
126, 11mp3an3 1449 . . . . 5 ((𝐴𝑆𝐵𝑇) → ∃𝑥𝑆𝑦𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦))
13 xpexg 7769 . . . . . 6 ((𝐴𝑆𝐵𝑇) → (𝐴 × 𝐵) ∈ V)
14 eqid 2735 . . . . . . 7 (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
1514elrnmpog 7568 . . . . . 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 257 . . . 4 ((𝐴𝑆𝐵𝑇) → (𝐴 × 𝐵) ∈ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
1817adantl 481 . . 3 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
195, 18sseldd 3996 . 2 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
201sxval 34171 . . 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 2842 1 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ (𝑆 ×s 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  wss 3963   × cxp 5687  ran crn 5690  cfv 6563  (class class class)co 7431  cmpo 7433  sigaGencsigagen 34119   ×s csx 34169
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-int 4952  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-siga 34090  df-sigagen 34120  df-sx 34170
This theorem is referenced by:  1stmbfm  34242  2ndmbfm  34243
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