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Theorem sxval 30031
 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 3198 . . 3 (𝑆𝑉𝑆 ∈ V)
2 elex 3198 . . 3 (𝑇𝑊𝑇 ∈ V)
3 id 22 . . . . . . 7 (𝑠 = 𝑆𝑠 = 𝑆)
4 eqidd 2622 . . . . . . 7 (𝑠 = 𝑆𝑡 = 𝑡)
5 eqidd 2622 . . . . . . 7 (𝑠 = 𝑆 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
63, 4, 5mpt2eq123dv 6670 . . . . . 6 (𝑠 = 𝑆 → (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)))
76rneqd 5313 . . . . 5 (𝑠 = 𝑆 → ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)))
87fveq2d 6152 . . . 4 (𝑠 = 𝑆 → (sigaGen‘ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦))))
9 eqidd 2622 . . . . . . 7 (𝑡 = 𝑇𝑆 = 𝑆)
10 id 22 . . . . . . 7 (𝑡 = 𝑇𝑡 = 𝑇)
11 eqidd 2622 . . . . . . 7 (𝑡 = 𝑇 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
129, 10, 11mpt2eq123dv 6670 . . . . . 6 (𝑡 = 𝑇 → (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
1312rneqd 5313 . . . . 5 (𝑡 = 𝑇 → ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
1413fveq2d 6152 . . . 4 (𝑡 = 𝑇 → (sigaGen‘ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
15 df-sx 30030 . . . 4 ×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦))))
16 fvex 6158 . . . 4 (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))) ∈ V
178, 14, 15, 16ovmpt2 6749 . . 3 ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
181, 2, 17syl2an 494 . 2 ((𝑆𝑉𝑇𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
19 sxval.1 . . 3 𝐴 = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
2019fveq2i 6151 . 2 (sigaGen‘𝐴) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
2118, 20syl6eqr 2673 1 ((𝑆𝑉𝑇𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186   × cxp 5072  ran crn 5075  ‘cfv 5847  (class class class)co 6604   ↦ cmpt2 6606  sigaGencsigagen 29979   ×s csx 30029 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-sx 30030 This theorem is referenced by:  sxsiga  30032  sxsigon  30033  elsx  30035  mbfmco2  30105  sxbrsigalem5  30128  sxbrsiga  30130
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