Theorem elsx 34158

Description: The cartesian product of two open sets is an element of the product sigma-algebra. (Contributed by Thierry Arnoux, 3-Jun-2017.)

Assertion

Ref	Expression
elsx	⊢ (((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐴 × 𝐵) ∈ (𝑆 ×s 𝑇))

Proof of Theorem elsx

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2740	. . . . . 6 ⊢ ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))
2	1	txbasex 23595	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
3		sssigagen 34109	. . . . 5 ⊢ (ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)) ∈ V → ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)) ⊆ (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
4	2, 3	syl 17	. . . 4 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)) ⊆ (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
5	4	adantr 480	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)) ⊆ (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
6		eqid 2740	. . . . . 6 ⊢ (𝐴 × 𝐵) = (𝐴 × 𝐵)
7		xpeq1 5714	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
8	7	eqeq2d 2751	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐴 × 𝐵) = (𝑥 × 𝑦) ↔ (𝐴 × 𝐵) = (𝐴 × 𝑦)))
9		xpeq2 5721	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
10	9	eqeq2d 2751	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 × 𝐵) = (𝐴 × 𝑦) ↔ (𝐴 × 𝐵) = (𝐴 × 𝐵)))
11	8, 10	rspc2ev 3648	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇 ∧ (𝐴 × 𝐵) = (𝐴 × 𝐵)) → ∃𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦))
12	6, 11	mp3an3 1450	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → ∃𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦))
13		xpexg 7785	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (𝐴 × 𝐵) ∈ V)
14		eqid 2740	. . . . . . 7 ⊢ (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))
15	14	elrnmpog 7585	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ V → ((𝐴 × 𝐵) ∈ ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦)))
16	13, 15	syl 17	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → ((𝐴 × 𝐵) ∈ ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)) ↔ ∃𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦)))
17	12, 16	mpbird 257	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (𝐴 × 𝐵) ∈ ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
18	17	adantl 481	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐴 × 𝐵) ∈ ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
19	5, 18	sseldd 4009	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐴 × 𝐵) ∈ (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
20	1	sxval 34154	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
21	20	adantr 480	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
22	19, 21	eleqtrrd 2847	1 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐴 × 𝐵) ∈ (𝑆 ×s 𝑇))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976   × cxp 5698  ran crn 5701  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  sigaGencsigagen 34102   ×_s csx 34152

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-siga 34073  df-sigagen 34103  df-sx 34153

This theorem is referenced by:  1stmbfm  34225  2ndmbfm  34226