Theorem sxval 34645

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 3478	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
2		elex 3478	. . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V)
3		id 23	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
4		eqidd 2766	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝑡 = 𝑡)
5		eqidd 2766	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
6	3, 4, 5	mpoeq123dv 7494	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)))
7	6	rneqd 5930	. . . . 5 ⊢ (𝑠 = 𝑆 → ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)))
8	7	fveq2d 6889	. . . 4 ⊢ (𝑠 = 𝑆 → (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))))
9		eqidd 2766	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝑆 = 𝑆)
10		id 23	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇)
11		eqidd 2766	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
12	9, 10, 11	mpoeq123dv 7494	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
13	12	rneqd 5930	. . . . 5 ⊢ (𝑡 = 𝑇 → ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
14	13	fveq2d 6889	. . . 4 ⊢ (𝑡 = 𝑇 → (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
15		df-sx 34644	. . . 4 ⊢ ×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))))
16		fvex 6898	. . . 4 ⊢ (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))) ∈ V
17	8, 14, 15, 16	ovmpo 7579	. . 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 6888	. 2 ⊢ (sigaGen‘𝐴) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
21	18, 20	eqtr4di 2818	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457   × cxp 5661  ran crn 5664  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  sigaGencsigagen 34593   ×_s csx 34643

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-sx 34644

This theorem is used by:  sxsiga  34646  sxsigon  34647  elsx  34649  mbfmco2  34720  sxbrsigalem5  34743  sxbrsiga  34745