Theorem sxval 34180

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 3468	. . 3 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
2		elex 3468	. . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V)
3		id 22	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
4		eqidd 2730	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝑡 = 𝑡)
5		eqidd 2730	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
6	3, 4, 5	mpoeq123dv 7464	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)))
7	6	rneqd 5902	. . . . 5 ⊢ (𝑠 = 𝑆 → ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)))
8	7	fveq2d 6862	. . . 4 ⊢ (𝑠 = 𝑆 → (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))))
9		eqidd 2730	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝑆 = 𝑆)
10		id 22	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝑡 = 𝑇)
11		eqidd 2730	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑥 × 𝑦) = (𝑥 × 𝑦))
12	9, 10, 11	mpoeq123dv 7464	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
13	12	rneqd 5902	. . . . 5 ⊢ (𝑡 = 𝑇 → ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
14	13	fveq2d 6862	. . . 4 ⊢ (𝑡 = 𝑇 → (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
15		df-sx 34179	. . . 4 ⊢ ×s = (𝑠 ∈ V, 𝑡 ∈ V ↦ (sigaGen‘ran (𝑥 ∈ 𝑠, 𝑦 ∈ 𝑡 ↦ (𝑥 × 𝑦))))
16		fvex 6871	. . . 4 ⊢ (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))) ∈ V
17	8, 14, 15, 16	ovmpo 7549	. . 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 6861	. 2 ⊢ (sigaGen‘𝐴) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
21	18, 20	eqtr4di 2782	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447   × cxp 5636  ran crn 5639  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  sigaGencsigagen 34128   ×_s csx 34178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-sx 34179

This theorem is referenced by:  sxsiga  34181  sxsigon  34182  elsx  34184  mbfmco2  34256  sxbrsigalem5  34279  sxbrsiga  34281