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Theorem ssrab2f 38787
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 3111 . . 3 𝑥{𝑥𝐴𝜑}
2 ssrab2f.1 . . 3 𝑥𝐴
31, 2dfss3f 3575 . 2 ({𝑥𝐴𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥𝐴𝜑}𝑥𝐴)
4 rabidim1 38774 . 2 (𝑥 ∈ {𝑥𝐴𝜑} → 𝑥𝐴)
53, 4mprgbir 2922 1 {𝑥𝐴𝜑} ⊆ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  wnfc 2748  {crab 2911  wss 3555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-in 3562  df-ss 3569
This theorem is referenced by:  dmmptssf  38913  mptssid  38925  fnlimfvre  39310  limsupequzmpt2  39354  smflimlem2  40287  smflim  40292  smfpimcclem  40320  smfsupxr  40329
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