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Theorem ssrab2f 45127
Description: Subclass relation for a restricted class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
ssrab2f.1 𝑥𝐴
Assertion
Ref Expression
ssrab2f {𝑥𝐴𝜑} ⊆ 𝐴

Proof of Theorem ssrab2f
StepHypRef Expression
1 nfrab1 3456 . . 3 𝑥{𝑥𝐴𝜑}
2 ssrab2f.1 . . 3 𝑥𝐴
31, 2dfss3f 3974 . 2 ({𝑥𝐴𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥𝐴𝜑}𝑥𝐴)
4 rabidim1 3458 . 2 (𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)
53, 4mprgbir 3067 1 {𝑥𝐴𝜑} ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  wnfc 2889  {crab 3435  wss 3950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rab 3436  df-ss 3967
This theorem is referenced by:  dmmptssf  45242  mptssid  45252  fnlimfvre  45694  limsupequzmpt2  45738  liminfequzmpt2  45811  pimltpnff  46723  pimgtmnff  46742  smflimlem2  46792  smflim  46797  smfpimcclem  46827  smfsupxr  46836  smfpimne2  46860
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