Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  unirestss Structured version   Visualization version   GIF version

Theorem unirestss 41267
Description: The union of an elementwise intersection is a subset of the underlying set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
unirestss.1 (𝜑𝐴𝑉)
unirestss.2 (𝜑𝐵𝑊)
Assertion
Ref Expression
unirestss (𝜑 (𝐴t 𝐵) ⊆ 𝐴)

Proof of Theorem unirestss
StepHypRef Expression
1 unirestss.1 . . 3 (𝜑𝐴𝑉)
2 unirestss.2 . . 3 (𝜑𝐵𝑊)
31, 2restuni6 41265 . 2 (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))
4 inss1 4202 . 2 ( 𝐴𝐵) ⊆ 𝐴
53, 4eqsstrdi 4018 1 (𝜑 (𝐴t 𝐵) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cin 3932  wss 3933   cuni 4830  (class class class)co 7145  t crest 16682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-rest 16684
This theorem is referenced by:  cnfsmf  42894
  Copyright terms: Public domain W3C validator