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Theorem elsx 32793
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 2736 . . . . . 6 ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
21txbasex 22917 . . . . 5 ((𝑆𝑉𝑇𝑊) → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
3 sssigagen 32744 . . . . 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 481 . . 3 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) ⊆ (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
6 eqid 2736 . . . . . 6 (𝐴 × 𝐵) = (𝐴 × 𝐵)
7 xpeq1 5647 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
87eqeq2d 2747 . . . . . . 7 (𝑥 = 𝐴 → ((𝐴 × 𝐵) = (𝑥 × 𝑦) ↔ (𝐴 × 𝐵) = (𝐴 × 𝑦)))
9 xpeq2 5654 . . . . . . . 8 (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
109eqeq2d 2747 . . . . . . 7 (𝑦 = 𝐵 → ((𝐴 × 𝐵) = (𝐴 × 𝑦) ↔ (𝐴 × 𝐵) = (𝐴 × 𝐵)))
118, 10rspc2ev 3592 . . . . . 6 ((𝐴𝑆𝐵𝑇 ∧ (𝐴 × 𝐵) = (𝐴 × 𝐵)) → ∃𝑥𝑆𝑦𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦))
126, 11mp3an3 1450 . . . . 5 ((𝐴𝑆𝐵𝑇) → ∃𝑥𝑆𝑦𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦))
13 xpexg 7684 . . . . . 6 ((𝐴𝑆𝐵𝑇) → (𝐴 × 𝐵) ∈ V)
14 eqid 2736 . . . . . . 7 (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
1514elrnmpog 7491 . . . . . 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 482 . . 3 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
195, 18sseldd 3945 . 2 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
201sxval 32789 . . 3 ((𝑆𝑉𝑇𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
2120adantr 481 . 2 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
2219, 21eleqtrrd 2841 1 (((𝑆𝑉𝑇𝑊) ∧ (𝐴𝑆𝐵𝑇)) → (𝐴 × 𝐵) ∈ (𝑆 ×s 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  wss 3910   × cxp 5631  ran crn 5634  cfv 6496  (class class class)co 7357  cmpo 7359  sigaGencsigagen 32737   ×s csx 32787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-siga 32708  df-sigagen 32738  df-sx 32788
This theorem is referenced by:  1stmbfm  32860  2ndmbfm  32861
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