Theorem ssrab2f 42619

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 3315	. . 3 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 𝜑}
2		ssrab2f.1	. . 3 ⊢ Ⅎ𝑥𝐴
3	1, 2	dfss3f 3916	. 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}𝑥 ∈ 𝐴)
4		rabidim1 3310	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → 𝑥 ∈ 𝐴)
5	3, 4	mprgbir 3080	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴

Syntax hints:   ∈ wcel 2109  Ⅎwnfc 2888  {crab 3069   ⊆ wss 3891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1544  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rab 3074  df-v 3432  df-in 3898  df-ss 3908

This theorem is referenced by:  dmmptssf  42728  mptssid  42738  fnlimfvre  43169  limsupequzmpt2  43213  liminfequzmpt2  43286  smflimlem2  44258  smflim  44263  smfpimcclem  44291  smfsupxr  44300