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Theorem ssrab2f 43791
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 3451 . . 3 𝑥{𝑥𝐴𝜑}
2 ssrab2f.1 . . 3 𝑥𝐴
31, 2dfss3f 3972 . 2 ({𝑥𝐴𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥𝐴𝜑}𝑥𝐴)
4 rabidim1 3453 . 2 (𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)
53, 4mprgbir 3068 1 {𝑥𝐴𝜑} ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  wnfc 2883  {crab 3432  wss 3947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rab 3433  df-v 3476  df-in 3954  df-ss 3964
This theorem is referenced by:  dmmptssf  43919  mptssid  43929  fnlimfvre  44376  limsupequzmpt2  44420  liminfequzmpt2  44493  pimltpnff  45405  pimgtmnff  45424  smflimlem2  45474  smflim  45479  smfpimcclem  45509  smfsupxr  45518  smfpimne2  45542
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