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Theorem restuni4 41378
 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 4176 . . 3 (𝐵 𝐴) = ( 𝐴𝐵)
21a1i 11 . 2 (𝜑 → (𝐵 𝐴) = ( 𝐴𝐵))
3 restuni4.2 . . 3 (𝜑𝐵 𝐴)
4 dfss 3951 . . 3 (𝐵 𝐴𝐵 = (𝐵 𝐴))
53, 4sylib 220 . 2 (𝜑𝐵 = (𝐵 𝐴))
6 restuni4.1 . . 3 (𝜑𝐴𝑉)
76uniexd 7460 . . . 4 (𝜑 𝐴 ∈ V)
87, 3ssexd 5219 . . 3 (𝜑𝐵 ∈ V)
96, 8restuni3 41375 . 2 (𝜑 (𝐴t 𝐵) = ( 𝐴𝐵))
102, 5, 93eqtr4rd 2865 1 (𝜑 (𝐴t 𝐵) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1531   ∈ wcel 2108  Vcvv 3493   ∩ cin 3933   ⊆ wss 3934  ∪ cuni 4830  (class class class)co 7148   ↾t crest 16686 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  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 7151  df-oprab 7152  df-mpo 7153  df-rest 16688 This theorem is referenced by:  restuni6  41379  restuni5  41380  subsaluni  42634  issmflelem  43012  smfpimltxr  43015  issmfgtlem  43023  issmfgt  43024  issmfgelem  43036  smfpimgtxr  43047  smfresal  43054
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