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Theorem sxval 34191
Description: Value of the product sigma-algebra operation. (Contributed by Thierry Arnoux, 1-Jun-2017.)
Hypothesis
Ref Expression
sxval.1 𝐴 = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
Assertion
Ref Expression
sxval ((𝑆𝑉𝑇𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘𝐴))
Distinct variable groups:   𝑥,𝑦,𝑆   𝑥,𝑇,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem sxval
Dummy variables 𝑡 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3501 . . 3 (𝑆𝑉𝑆 ∈ V)
2 elex 3501 . . 3 (𝑇𝑊𝑇 ∈ V)
3 id 22 . . . . . . 7 (𝑠 = 𝑆𝑠 = 𝑆)
4 eqidd 2738 . . . . . . 7 (𝑠 = 𝑆𝑡 = 𝑡)
5 eqidd 2738 . . . . . . 7 (𝑠 = 𝑆 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
63, 4, 5mpoeq123dv 7508 . . . . . 6 (𝑠 = 𝑆 → (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)))
76rneqd 5949 . . . . 5 (𝑠 = 𝑆 → ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)))
87fveq2d 6910 . . . 4 (𝑠 = 𝑆 → (sigaGen‘ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦))))
9 eqidd 2738 . . . . . . 7 (𝑡 = 𝑇𝑆 = 𝑆)
10 id 22 . . . . . . 7 (𝑡 = 𝑇𝑡 = 𝑇)
11 eqidd 2738 . . . . . . 7 (𝑡 = 𝑇 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
129, 10, 11mpoeq123dv 7508 . . . . . 6 (𝑡 = 𝑇 → (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
1312rneqd 5949 . . . . 5 (𝑡 = 𝑇 → ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
1413fveq2d 6910 . . . 4 (𝑡 = 𝑇 → (sigaGen‘ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
15 df-sx 34190 . . . 4 ×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦))))
16 fvex 6919 . . . 4 (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))) ∈ V
178, 14, 15, 16ovmpo 7593 . . 3 ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
181, 2, 17syl2an 596 . 2 ((𝑆𝑉𝑇𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
19 sxval.1 . . 3 𝐴 = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
2019fveq2i 6909 . 2 (sigaGen‘𝐴) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
2118, 20eqtr4di 2795 1 ((𝑆𝑉𝑇𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480   × cxp 5683  ran crn 5686  cfv 6561  (class class class)co 7431  cmpo 7433  sigaGencsigagen 34139   ×s csx 34189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sx 34190
This theorem is referenced by:  sxsiga  34192  sxsigon  34193  elsx  34195  mbfmco2  34267  sxbrsigalem5  34290  sxbrsiga  34292
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