Theorem elsx 34027

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 2726	. . . . . 6 ⊢ ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))
2	1	txbasex 23561	. . . . 5 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)) ∈ V)
3		sssigagen 33978	. . . . 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 479	. . 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 2726	. . . . . 6 ⊢ (𝐴 × 𝐵) = (𝐴 × 𝐵)
7		xpeq1 5696	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 × 𝑦) = (𝐴 × 𝑦))
8	7	eqeq2d 2737	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝐴 × 𝐵) = (𝑥 × 𝑦) ↔ (𝐴 × 𝐵) = (𝐴 × 𝑦)))
9		xpeq2 5703	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐴 × 𝑦) = (𝐴 × 𝐵))
10	9	eqeq2d 2737	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 × 𝐵) = (𝐴 × 𝑦) ↔ (𝐴 × 𝐵) = (𝐴 × 𝐵)))
11	8, 10	rspc2ev 3621	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇 ∧ (𝐴 × 𝐵) = (𝐴 × 𝐵)) → ∃𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦))
12	6, 11	mp3an3 1447	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → ∃𝑥 ∈ 𝑆 ∃𝑦 ∈ 𝑇 (𝐴 × 𝐵) = (𝑥 × 𝑦))
13		xpexg 7758	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (𝐴 × 𝐵) ∈ V)
14		eqid 2726	. . . . . . 7 ⊢ (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))
15	14	elrnmpog 7561	. . . . . 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 256	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇) → (𝐴 × 𝐵) ∈ ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
18	17	adantl 480	. . 3 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐴 × 𝐵) ∈ ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦)))
19	5, 18	sseldd 3980	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐴 × 𝐵) ∈ (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
20	1	sxval 34023	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
21	20	adantr 479	. 2 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑇 ↦ (𝑥 × 𝑦))))
22	19, 21	eleqtrrd 2829	1 ⊢ (((𝑆 ∈ 𝑉 ∧ 𝑇 ∈ 𝑊) ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐴 × 𝐵) ∈ (𝑆 ×s 𝑇))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∃wrex 3060  Vcvv 3462   ⊆ wss 3947   × cxp 5680  ran crn 5683  ‘cfv 6554  (class class class)co 7424   ∈ cmpo 7426  sigaGencsigagen 33971   ×_s csx 34021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-siga 33942  df-sigagen 33972  df-sx 34022

This theorem is referenced by:  1stmbfm  34094  2ndmbfm  34095