Theorem restuni4 45131

Description: The underlying set of a subspace induced by the ↾_t operator. The result can be applied, for instance, to topologies and sigma-algebras. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
restuni4.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
restuni4.2	⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴)

Assertion

Ref	Expression
restuni4	⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵)

Proof of Theorem restuni4

Step	Hyp	Ref	Expression
1		incom 4208	. . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)
2	1	a1i 11	. 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵))
3		restuni4.2	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴)
4		dfss 3969	. . 3 ⊢ (𝐵 ⊆ ∪ 𝐴 ↔ 𝐵 = (𝐵 ∩ ∪ 𝐴))
5	3, 4	sylib 218	. 2 ⊢ (𝜑 → 𝐵 = (𝐵 ∩ ∪ 𝐴))
6		restuni4.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7	6	uniexd 7763	. . . 4 ⊢ (𝜑 → ∪ 𝐴 ∈ V)
8	7, 3	ssexd 5323	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
9	6, 8	restuni3 45128	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵))
10	2, 5, 9	3eqtr4rd 2787	1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  Vcvv 3479   ∩ cin 3949   ⊆ wss 3950  ∪ cuni 4906  (class class class)co 7432   ↾_t crest 17466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-rest 17468

This theorem is referenced by:  restuni6  45132  restuni5  45133  subsaluni  46380  issmflelem  46764  issmfgtlem  46775  issmfgt  46776  issmfgelem  46789  smfresal  46808