Theorem sxval 34195

Description: Value of the product sigma-algebra operation. (Contributed by Thierry Arnoux, 1-Jun-2017.)

Hypothesis

Ref	Expression
sxval.1	⊢ 𝐴 = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))

Assertion

Ref	Expression
sxval	⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (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.

Step	Hyp	Ref	Expression
1		elex 3457	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
2		elex 3457	. . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V)
3		id 22	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
4		eqidd 2732	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝑡 = 𝑡)
5		eqidd 2732	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
6	3, 4, 5	mpoeq123dv 7416	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)))
7	6	rneqd 5873	. . . . 5 ⊢ (𝑠 = 𝑆 → ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)))
8	7	fveq2d 6821	. . . 4 ⊢ (𝑠 = 𝑆 → (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))))
9		eqidd 2732	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝑆 = 𝑆)
10		id 22	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇)
11		eqidd 2732	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
12	9, 10, 11	mpoeq123dv 7416	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
13	12	rneqd 5873	. . . . 5 ⊢ (𝑡 = 𝑇 → ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
14	13	fveq2d 6821	. . . 4 ⊢ (𝑡 = 𝑇 → (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
15		df-sx 34194	. . . 4 ⊢ ×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))))
16		fvex 6830	. . . 4 ⊢ (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))) ∈ V
17	8, 14, 15, 16	ovmpo 7501	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
18	1, 2, 17	syl2an 596	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
19		sxval.1	. . 3 ⊢ 𝐴 = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))
20	19	fveq2i 6820	. 2 ⊢ (sigaGen‘𝐴) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
21	18, 20	eqtr4di 2784	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   × cxp 5609  ran crn 5612  ‘cfv 6476  (class class class)co 7341   ∈ cmpo 7343  sigaGencsigagen 34143   ×_s csx 34193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-iota 6432  df-fun 6478  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-sx 34194

This theorem is referenced by:  sxsiga  34196  sxsigon  34197  elsx  34199  mbfmco2  34270  sxbrsigalem5  34293  sxbrsiga  34295