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Theorem restuni4 44385
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 4196 . . 3 (𝐵 𝐴) = ( 𝐴𝐵)
21a1i 11 . 2 (𝜑 → (𝐵 𝐴) = ( 𝐴𝐵))
3 restuni4.2 . . 3 (𝜑𝐵 𝐴)
4 dfss 3961 . . 3 (𝐵 𝐴𝐵 = (𝐵 𝐴))
53, 4sylib 217 . 2 (𝜑𝐵 = (𝐵 𝐴))
6 restuni4.1 . . 3 (𝜑𝐴𝑉)
76uniexd 7729 . . . 4 (𝜑 𝐴 ∈ V)
87, 3ssexd 5317 . . 3 (𝜑𝐵 ∈ V)
96, 8restuni3 44382 . 2 (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))
102, 5, 93eqtr4rd 2777 1 (𝜑 (𝐴t 𝐵) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3468  cin 3942  wss 3943   cuni 4902  (class class class)co 7405  t crest 17375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-rest 17377
This theorem is referenced by:  restuni6  44386  restuni5  44387  subsaluni  45648  issmflelem  46032  issmfgtlem  46043  issmfgt  46044  issmfgelem  46057  smfresal  46076
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