Theorem restuni4 38759

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 3788	. . 3 ⊢ (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵)
2	1	a1i 11	. 2 ⊢ (𝜑 → (𝐵 ∩ ∪ 𝐴) = (∪ 𝐴 ∩ 𝐵))
3		restuni4.2	. . 3 ⊢ (𝜑 → 𝐵 ⊆ ∪ 𝐴)
4		dfss 3575	. . 3 ⊢ (𝐵 ⊆ ∪ 𝐴 ↔ 𝐵 = (𝐵 ∩ ∪ 𝐴))
5	3, 4	sylib 208	. 2 ⊢ (𝜑 → 𝐵 = (𝐵 ∩ ∪ 𝐴))
6		restuni4.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7	6	uniexd 38733	. . . 4 ⊢ (𝜑 → ∪ 𝐴 ∈ V)
8	7, 3	ssexd 4770	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
9	6, 8	restuni3 38756	. 2 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = (∪ 𝐴 ∩ 𝐵))
10	2, 5, 9	3eqtr4rd 2671	1 ⊢ (𝜑 → ∪ (𝐴 ↾t 𝐵) = 𝐵)

Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1992  Vcvv 3191   ∩ cin 3559   ⊆ wss 3560  ∪ cuni 4407  (class class class)co 6605   ↾_t crest 15997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-rest 15999

This theorem is referenced by:  restuni6  38760  restuni5  38761  subsaluni  39853  issmflelem  40228  smfpimltxr  40231  issmfgtlem  40239  issmfgt  40240  issmfgelem  40252  smfpimgtxr  40263  smfresal  40270