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Theorem sxval 32601
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 3461 . . 3 (𝑆 ∈ 𝑉 β†’ 𝑆 ∈ V)
2 elex 3461 . . 3 (𝑇 ∈ π‘Š β†’ 𝑇 ∈ V)
3 id 22 . . . . . . 7 (𝑠 = 𝑆 β†’ 𝑠 = 𝑆)
4 eqidd 2738 . . . . . . 7 (𝑠 = 𝑆 β†’ 𝑑 = 𝑑)
5 eqidd 2738 . . . . . . 7 (𝑠 = 𝑆 β†’ (π‘₯ Γ— 𝑦) = (π‘₯ Γ— 𝑦))
63, 4, 5mpoeq123dv 7426 . . . . . 6 (𝑠 = 𝑆 β†’ (π‘₯ ∈ 𝑠, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦)) = (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦)))
76rneqd 5891 . . . . 5 (𝑠 = 𝑆 β†’ ran (π‘₯ ∈ 𝑠, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦)) = ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦)))
87fveq2d 6843 . . . 4 (𝑠 = 𝑆 β†’ (sigaGenβ€˜ran (π‘₯ ∈ 𝑠, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦))) = (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦))))
9 eqidd 2738 . . . . . . 7 (𝑑 = 𝑇 β†’ 𝑆 = 𝑆)
10 id 22 . . . . . . 7 (𝑑 = 𝑇 β†’ 𝑑 = 𝑇)
11 eqidd 2738 . . . . . . 7 (𝑑 = 𝑇 β†’ (π‘₯ Γ— 𝑦) = (π‘₯ Γ— 𝑦))
129, 10, 11mpoeq123dv 7426 . . . . . 6 (𝑑 = 𝑇 β†’ (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦)) = (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)))
1312rneqd 5891 . . . . 5 (𝑑 = 𝑇 β†’ ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦)) = ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)))
1413fveq2d 6843 . . . 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 32600 . . . 4 Γ—s = (𝑠 ∈ V, 𝑑 ∈ V ↦ (sigaGenβ€˜ran (π‘₯ ∈ 𝑠, 𝑦 ∈ 𝑑 ↦ (π‘₯ Γ— 𝑦))))
16 fvex 6852 . . . 4 (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))) ∈ V
178, 14, 15, 16ovmpo 7509 . . 3 ((𝑆 ∈ V ∧ 𝑇 ∈ V) β†’ (𝑆 Γ—s 𝑇) = (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦))))
181, 2, 17syl2an 596 . 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 6842 . 2 (sigaGenβ€˜π΄) = (sigaGenβ€˜ran (π‘₯ ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (π‘₯ Γ— 𝑦)))
2118, 20eqtr4di 2795 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 396   = wceq 1541   ∈ wcel 2106  Vcvv 3443   Γ— cxp 5629  ran crn 5632  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353  sigaGencsigagen 32549   Γ—s csx 32599
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-sx 32600
This theorem is referenced by:  sxsiga  32602  sxsigon  32603  elsx  32605  mbfmco2  32677  sxbrsigalem5  32700  sxbrsiga  32702
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