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Theorem sxval 30583
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 3352 . . 3 (𝑆𝑉𝑆 ∈ V)
2 elex 3352 . . 3 (𝑇𝑊𝑇 ∈ V)
3 id 22 . . . . . . 7 (𝑠 = 𝑆𝑠 = 𝑆)
4 eqidd 2761 . . . . . . 7 (𝑠 = 𝑆𝑡 = 𝑡)
5 eqidd 2761 . . . . . . 7 (𝑠 = 𝑆 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
63, 4, 5mpt2eq123dv 6883 . . . . . 6 (𝑠 = 𝑆 → (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)))
76rneqd 5508 . . . . 5 (𝑠 = 𝑆 → ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)))
87fveq2d 6357 . . . 4 (𝑠 = 𝑆 → (sigaGen‘ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦))))
9 eqidd 2761 . . . . . . 7 (𝑡 = 𝑇𝑆 = 𝑆)
10 id 22 . . . . . . 7 (𝑡 = 𝑇𝑡 = 𝑇)
11 eqidd 2761 . . . . . . 7 (𝑡 = 𝑇 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
129, 10, 11mpt2eq123dv 6883 . . . . . 6 (𝑡 = 𝑇 → (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
1312rneqd 5508 . . . . 5 (𝑡 = 𝑇 → ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
1413fveq2d 6357 . . . 4 (𝑡 = 𝑇 → (sigaGen‘ran (𝑥𝑆, 𝑦𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
15 df-sx 30582 . . . 4 ×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥𝑠, 𝑦𝑡 ↦ (𝑥 × 𝑦))))
16 fvex 6363 . . . 4 (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))) ∈ V
178, 14, 15, 16ovmpt2 6962 . . 3 ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
181, 2, 17syl2an 495 . 2 ((𝑆𝑉𝑇𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))))
19 sxval.1 . . 3 𝐴 = ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦))
2019fveq2i 6356 . 2 (sigaGen‘𝐴) = (sigaGen‘ran (𝑥𝑆, 𝑦𝑇 ↦ (𝑥 × 𝑦)))
2118, 20syl6eqr 2812 1 ((𝑆𝑉𝑇𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340   × cxp 5264  ran crn 5267  cfv 6049  (class class class)co 6814  cmpt2 6816  sigaGencsigagen 30531   ×s csx 30581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-sx 30582
This theorem is referenced by:  sxsiga  30584  sxsigon  30585  elsx  30587  mbfmco2  30657  sxbrsigalem5  30680  sxbrsiga  30682
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