Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  restuni4 Structured version   Visualization version   GIF version

Theorem restuni4 40118
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 (𝜑𝐴𝑉)
restuni4.2 (𝜑𝐵 𝐴)
Assertion
Ref Expression
restuni4 (𝜑 (𝐴t 𝐵) = 𝐵)

Proof of Theorem restuni4
StepHypRef Expression
1 incom 4034 . . 3 (𝐵 𝐴) = ( 𝐴𝐵)
21a1i 11 . 2 (𝜑 → (𝐵 𝐴) = ( 𝐴𝐵))
3 restuni4.2 . . 3 (𝜑𝐵 𝐴)
4 dfss 3813 . . 3 (𝐵 𝐴𝐵 = (𝐵 𝐴))
53, 4sylib 210 . 2 (𝜑𝐵 = (𝐵 𝐴))
6 restuni4.1 . . 3 (𝜑𝐴𝑉)
76uniexd 40097 . . . 4 (𝜑 𝐴 ∈ V)
87, 3ssexd 5032 . . 3 (𝜑𝐵 ∈ V)
96, 8restuni3 40115 . 2 (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))
102, 5, 93eqtr4rd 2872 1 (𝜑 (𝐴t 𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  Vcvv 3414  cin 3797  wss 3798   cuni 4660  (class class class)co 6910  t crest 16441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-rest 16443
This theorem is referenced by:  restuni6  40119  restuni5  40120  subsaluni  41367  issmflelem  41745  smfpimltxr  41748  issmfgtlem  41756  issmfgt  41757  issmfgelem  41769  smfpimgtxr  41780  smfresal  41787
  Copyright terms: Public domain W3C validator