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Theorem sxval 34171
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 3499 . . 3 (𝑆𝑉𝑆 ∈ V)
2 elex 3499 . . 3 (𝑇𝑊𝑇 ∈ V)
3 id 22 . . . . . . 7 (𝑠 = 𝑆𝑠 = 𝑆)
4 eqidd 2736 . . . . . . 7 (𝑠 = 𝑆𝑡 = 𝑡)
5 eqidd 2736 . . . . . . 7 (𝑠 = 𝑆 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
63, 4, 5mpoeq123dv 7508 . . . . . 6 (𝑠 = 𝑆 → (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)))
76rneqd 5952 . . . . 5 (𝑠 = 𝑆 → ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)))
87fveq2d 6911 . . . 4 (𝑠 = 𝑆 → (sigaGen‘ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦))))
9 eqidd 2736 . . . . . . 7 (𝑡 = 𝑇𝑆 = 𝑆)
10 id 22 . . . . . . 7 (𝑡 = 𝑇𝑡 = 𝑇)
11 eqidd 2736 . . . . . . 7 (𝑡 = 𝑇 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
129, 10, 11mpoeq123dv 7508 . . . . . 6 (𝑡 = 𝑇 → (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
1312rneqd 5952 . . . . 5 (𝑡 = 𝑇 → ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
1413fveq2d 6911 . . . 4 (𝑡 = 𝑇 → (sigaGen‘ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
15 df-sx 34170 . . . 4 ×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦))))
16 fvex 6920 . . . 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 6910 . 2 (sigaGen‘𝐴) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
2118, 20eqtr4di 2793 1 ((𝑆𝑉𝑇𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478   × 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-pr 5438
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-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sx 34170
This theorem is referenced by:  sxsiga  34172  sxsigon  34173  elsx  34175  mbfmco2  34247  sxbrsigalem5  34270  sxbrsiga  34272
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