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Theorem elsx 34027
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 2726 . . . . . 6 ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
21txbasex 23561 . . . . 5 ((𝑆𝑉𝑇𝑊) → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
3 sssigagen 33978 . . . . 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 479 . . 3 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ⊆ (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
6 eqid 2726 . . . . . 6 (𝐴 × 𝐵) = (𝐴 × 𝐵)
7 xpeq1 5696 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
87eqeq2d 2737 . . . . . . 7 (𝑥 = 𝐴 → ((𝐴 × 𝐵) = (𝑥 × 𝑦) ↔ (𝐴 × 𝐵) = (𝐴 × 𝑦)))
9 xpeq2 5703 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
109eqeq2d 2737 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 × 𝐵) = (𝐴 × 𝑦) ↔ (𝐴 × 𝐵) = (𝐴 × 𝐵)))
118, 10rspc2ev 3621 . . . . . 6 ((𝐴𝑆𝐵𝑇 ∧ (𝐴 × 𝐵) = (𝐴 × 𝐵)) → ∃𝑥𝑆𝑦𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦))
126, 11mp3an3 1447 . . . . 5 ((𝐴𝑆𝐵𝑇) → ∃𝑥𝑆𝑦𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦))
13 xpexg 7758 . . . . . 6 ((𝐴𝑆𝐵𝑇) → (𝐴 × 𝐵) ∈ V)
14 eqid 2726 . . . . . . 7 (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
1514elrnmpog 7561 . . . . . 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 256 . . . 4 ((𝐴𝑆𝐵𝑇) → (𝐴 × 𝐵) ∈ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
1817adantl 480 . . 3 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
195, 18sseldd 3980 . 2 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
201sxval 34023 . . 3 ((𝑆𝑉𝑇𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
2120adantr 479 . 2 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
2219, 21eleqtrrd 2829 1 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ (𝑆 ×s 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wrex 3060  Vcvv 3462  wss 3947   × cxp 5680  ran crn 5683  cfv 6554  (class class class)co 7424  cmpo 7426  sigaGencsigagen 33971   ×s csx 34021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-siga 33942  df-sigagen 33972  df-sx 34022
This theorem is referenced by:  1stmbfm  34094  2ndmbfm  34095
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