Theorem ssrab2f 45357

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 3419	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
2		ssrab2f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	1, 2	dfss3f 3925	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}𝑥 ∈ 𝐴)
4		rabidim1 3421	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)
5	3, 4	mprgbir 3058	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴

Syntax hints:   ∈ wcel 2113  Ⅎwnfc 2883  {crab 3399   ⊆ wss 3901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rab 3400  df-ss 3918

This theorem is referenced by:  dmmptssf  45472  mptssid  45481  fnlimfvre  45914  limsupequzmpt2  45958  liminfequzmpt2  46031  pimltpnff  46943  pimgtmnff  46962  smflimlem2  47012  smflim  47017  smfpimcclem  47047  smfsupxr  47056  smfpimne2  47080