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Theorem sxval 31559
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 3459 . . 3 (𝑆𝑉𝑆 ∈ V)
2 elex 3459 . . 3 (𝑇𝑊𝑇 ∈ V)
3 id 22 . . . . . . 7 (𝑠 = 𝑆𝑠 = 𝑆)
4 eqidd 2799 . . . . . . 7 (𝑠 = 𝑆𝑡 = 𝑡)
5 eqidd 2799 . . . . . . 7 (𝑠 = 𝑆 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
63, 4, 5mpoeq123dv 7208 . . . . . 6 (𝑠 = 𝑆 → (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)))
76rneqd 5772 . . . . 5 (𝑠 = 𝑆 → ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)))
87fveq2d 6649 . . . 4 (𝑠 = 𝑆 → (sigaGen‘ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦))))
9 eqidd 2799 . . . . . . 7 (𝑡 = 𝑇𝑆 = 𝑆)
10 id 22 . . . . . . 7 (𝑡 = 𝑇𝑡 = 𝑇)
11 eqidd 2799 . . . . . . 7 (𝑡 = 𝑇 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
129, 10, 11mpoeq123dv 7208 . . . . . 6 (𝑡 = 𝑇 → (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
1312rneqd 5772 . . . . 5 (𝑡 = 𝑇 → ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
1413fveq2d 6649 . . . 4 (𝑡 = 𝑇 → (sigaGen‘ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
15 df-sx 31558 . . . 4 ×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦))))
16 fvex 6658 . . . 4 (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))) ∈ V
178, 14, 15, 16ovmpo 7289 . . 3 ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
181, 2, 17syl2an 598 . 2 ((𝑆𝑉𝑇𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
19 sxval.1 . . 3 𝐴 = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
2019fveq2i 6648 . 2 (sigaGen‘𝐴) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
2118, 20eqtr4di 2851 1 ((𝑆𝑉𝑇𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  Vcvv 3441   × cxp 5517  ran crn 5520  cfv 6324  (class class class)co 7135  cmpo 7137  sigaGencsigagen 31507   ×s csx 31557
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-sx 31558
This theorem is referenced by:  sxsiga  31560  sxsigon  31561  elsx  31563  mbfmco2  31633  sxbrsigalem5  31656  sxbrsiga  31658
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