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Theorem elsx 29390
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 2609 . . . . . 6 ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
21txbasex 21121 . . . . 5 ((𝑆𝑉𝑇𝑊) → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
3 sssigagen 29341 . . . . 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 2609 . . . . . 6 (𝐴 × 𝐵) = (𝐴 × 𝐵)
7 xpeq1 5042 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
87eqeq2d 2619 . . . . . . 7 (𝑥 = 𝐴 → ((𝐴 × 𝐵) = (𝑥 × 𝑦) ↔ (𝐴 × 𝐵) = (𝐴 × 𝑦)))
9 xpeq2 5043 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
109eqeq2d 2619 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 × 𝐵) = (𝐴 × 𝑦) ↔ (𝐴 × 𝐵) = (𝐴 × 𝐵)))
118, 10rspc2ev 3294 . . . . . 6 ((𝐴𝑆𝐵𝑇 ∧ (𝐴 × 𝐵) = (𝐴 × 𝐵)) → ∃𝑥𝑆𝑦𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦))
126, 11mp3an3 1404 . . . . 5 ((𝐴𝑆𝐵𝑇) → ∃𝑥𝑆𝑦𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦))
13 xpexg 6835 . . . . . 6 ((𝐴𝑆𝐵𝑇) → (𝐴 × 𝐵) ∈ V)
14 eqid 2609 . . . . . . 7 (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
1514elrnmpt2g 6648 . . . . . 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 245 . . . 4 ((𝐴𝑆𝐵𝑇) → (𝐴 × 𝐵) ∈ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
1817adantl 480 . . 3 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
195, 18sseldd 3568 . 2 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
201sxval 29386 . . 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 2690 1 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ (𝑆 ×s 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wrex 2896  Vcvv 3172  wss 3539   × cxp 5026  ran crn 5029  cfv 5790  (class class class)co 6527  cmpt2 6529  sigaGencsigagen 29334   ×s csx 29384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-siga 29304  df-sigagen 29335  df-sx 29385
This theorem is referenced by:  1stmbfm  29455  2ndmbfm  29456
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