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Theorem sxval 33708
Description: Value of the product sigma-algebra operation. (Contributed by Thierry Arnoux, 1-Jun-2017.)
Hypothesis
Ref Expression
sxval.1 𝐴 = ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))
Assertion
Ref Expression
sxval ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ π‘Š) β†’ (𝑆 Γ—s 𝑇) = (sigaGenβ€˜π΄))
Distinct variable groups:   π‘₯,𝑦,𝑆   π‘₯,𝑇,𝑦
Allowed substitution hints:   𝐴(π‘₯,𝑦)   𝑉(π‘₯,𝑦)   π‘Š(π‘₯,𝑦)

Proof of Theorem sxval
Dummy variables 𝑑 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3485 . . 3 (𝑆 ∈ 𝑉 β†’ 𝑆 ∈ V)
2 elex 3485 . . 3 (𝑇 ∈ π‘Š β†’ 𝑇 ∈ V)
3 id 22 . . . . . . 7 (𝑠 = 𝑆 β†’ 𝑠 = 𝑆)
4 eqidd 2725 . . . . . . 7 (𝑠 = 𝑆 β†’ 𝑑 = 𝑑)
5 eqidd 2725 . . . . . . 7 (𝑠 = 𝑆 β†’ (π‘₯ Γ— 𝑦) = (π‘₯ Γ— 𝑦))
63, 4, 5mpoeq123dv 7477 . . . . . 6 (𝑠 = 𝑆 β†’ (π‘₯ ∈ 𝑠, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦)) = (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦)))
76rneqd 5928 . . . . 5 (𝑠 = 𝑆 β†’ ran (π‘₯ ∈ 𝑠, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦)) = ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦)))
87fveq2d 6886 . . . 4 (𝑠 = 𝑆 β†’ (sigaGenβ€˜ran (π‘₯ ∈ 𝑠, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦))) = (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦))))
9 eqidd 2725 . . . . . . 7 (𝑑 = 𝑇 β†’ 𝑆 = 𝑆)
10 id 22 . . . . . . 7 (𝑑 = 𝑇 β†’ 𝑑 = 𝑇)
11 eqidd 2725 . . . . . . 7 (𝑑 = 𝑇 β†’ (π‘₯ Γ— 𝑦) = (π‘₯ Γ— 𝑦))
129, 10, 11mpoeq123dv 7477 . . . . . 6 (𝑑 = 𝑇 β†’ (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦)) = (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)))
1312rneqd 5928 . . . . 5 (𝑑 = 𝑇 β†’ ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦)) = ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)))
1413fveq2d 6886 . . . 4 (𝑑 = 𝑇 β†’ (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦))) = (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))))
15 df-sx 33707 . . . 4 Γ—s = (𝑠 ∈ V, 𝑑 ∈ V ↦ (sigaGenβ€˜ran (π‘₯ ∈ 𝑠, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦))))
16 fvex 6895 . . . 4 (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))) ∈ V
178, 14, 15, 16ovmpo 7561 . . 3 ((𝑆 ∈ V ∧ 𝑇 ∈ V) β†’ (𝑆 Γ—s 𝑇) = (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))))
181, 2, 17syl2an 595 . 2 ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ π‘Š) β†’ (𝑆 Γ—s 𝑇) = (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))))
19 sxval.1 . . 3 𝐴 = ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))
2019fveq2i 6885 . 2 (sigaGenβ€˜π΄) = (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)))
2118, 20eqtr4di 2782 1 ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ π‘Š) β†’ (𝑆 Γ—s 𝑇) = (sigaGenβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466   Γ— cxp 5665  ran crn 5668  β€˜cfv 6534  (class class class)co 7402   ∈ cmpo 7404  sigaGencsigagen 33656   Γ—s csx 33706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-sx 33707
This theorem is referenced by:  sxsiga  33709  sxsigon  33710  elsx  33712  mbfmco2  33784  sxbrsigalem5  33807  sxbrsiga  33809
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