Theorem ssrab2f 45571

Description: Subclass relation for a restricted class. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypothesis

Ref	Expression
ssrab2f.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
ssrab2f	⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴

Proof of Theorem ssrab2f

Step	Hyp	Ref	Expression
1		nfrab1 3412	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
2		ssrab2f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	1, 2	dfss3f 3914	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}𝑥 ∈ 𝐴)
4		rabidim1 3414	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)
5	3, 4	mprgbir 3061	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴

Syntax hints:   ∈ wcel 2119  Ⅎwnfc 2887  {crab 3392   ⊆ wss 3890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rab 3393  df-ss 3907

This theorem is referenced by:  dmmptssf  45683  mptssid  45692  fnlimfvre  46124  limsupequzmpt2  46168  liminfequzmpt2  46241  pimltpnff  47153  pimgtmnff  47172  smflimlem2  47222  smflim  47227  smfpimcclem  47257  smfsupxr  47266  smfpimne2  47290