Theorem ssrab2f 45019

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 3464	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
2		ssrab2f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	1, 2	dfss3f 4000	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}𝑥 ∈ 𝐴)
4		rabidim1 3466	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)
5	3, 4	mprgbir 3074	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴

Syntax hints:   ∈ wcel 2108  Ⅎwnfc 2893  {crab 3443   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rab 3444  df-ss 3993

This theorem is referenced by:  dmmptssf  45139  mptssid  45149  fnlimfvre  45595  limsupequzmpt2  45639  liminfequzmpt2  45712  pimltpnff  46624  pimgtmnff  46643  smflimlem2  46693  smflim  46698  smfpimcclem  46728  smfsupxr  46737  smfpimne2  46761