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Theorem ssrab2f 45057
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 3454 . . 3 𝑥{𝑥𝐴𝜑}
2 ssrab2f.1 . . 3 𝑥𝐴
31, 2dfss3f 3987 . 2 ({𝑥𝐴𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥𝐴𝜑}𝑥𝐴)
4 rabidim1 3456 . 2 (𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)
53, 4mprgbir 3066 1 {𝑥𝐴𝜑} ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  wnfc 2888  {crab 3433  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-ss 3980
This theorem is referenced by:  dmmptssf  45175  mptssid  45185  fnlimfvre  45630  limsupequzmpt2  45674  liminfequzmpt2  45747  pimltpnff  46659  pimgtmnff  46678  smflimlem2  46728  smflim  46733  smfpimcclem  46763  smfsupxr  46772  smfpimne2  46796
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