Theorem sxval 34701

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 3471	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
2		elex 3471	. . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V)
3		id 23	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
4		eqidd 2761	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝑡 = 𝑡)
5		eqidd 2761	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
6	3, 4, 5	mpoeq123dv 7488	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)))
7	6	rneqd 5922	. . . . 5 ⊢ (𝑠 = 𝑆 → ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)))
8	7	fveq2d 6882	. . . 4 ⊢ (𝑠 = 𝑆 → (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))))
9		eqidd 2761	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝑆 = 𝑆)
10		id 23	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇)
11		eqidd 2761	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
12	9, 10, 11	mpoeq123dv 7488	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
13	12	rneqd 5922	. . . . 5 ⊢ (𝑡 = 𝑇 → ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
14	13	fveq2d 6882	. . . 4 ⊢ (𝑡 = 𝑇 → (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
15		df-sx 34700	. . . 4 ⊢ ×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))))
16		fvex 6891	. . . 4 ⊢ (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))) ∈ V
17	8, 14, 15, 16	ovmpo 7573	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑇 ∈ V) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
18	1, 2, 17	syl2an 608	. 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 6881	. 2 ⊢ (sigaGen‘𝐴) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
21	18, 20	eqtr4di 2813	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450   × cxp 5653  ran crn 5656  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  sigaGencsigagen 34649   ×_s csx 34699

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-sx 34700

This theorem is used by:  sxsiga  34702  sxsigon  34703  elsx  34705  mbfmco2  34776  sxbrsigalem5  34799  sxbrsiga  34801