MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabssrabd Structured version   Visualization version   GIF version

Theorem rabssrabd 4016
Description: Subclass of a restricted class abstraction. (Contributed by AV, 4-Jun-2022.)
Hypotheses
Ref Expression
rabssrabd.1 (𝜑𝐴𝐵)
rabssrabd.2 ((𝜑𝜓𝑥𝐴) → 𝜒)
Assertion
Ref Expression
rabssrabd (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜒})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem rabssrabd
StepHypRef Expression
1 3anan32 1096 . . . . 5 ((𝜑𝜓𝑥𝐴) ↔ ((𝜑𝑥𝐴) ∧ 𝜓))
2 rabssrabd.2 . . . . 5 ((𝜑𝜓𝑥𝐴) → 𝜒)
31, 2sylbir 234 . . . 4 (((𝜑𝑥𝐴) ∧ 𝜓) → 𝜒)
43ex 413 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
54ss2rabdv 4009 . 2 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
6 rabssrabd.1 . . 3 (𝜑𝐴𝐵)
7 rabss2 4011 . . 3 (𝐴𝐵 → {𝑥𝐴𝜒} ⊆ {𝑥𝐵𝜒})
86, 7syl 17 . 2 (𝜑 → {𝑥𝐴𝜒} ⊆ {𝑥𝐵𝜒})
95, 8sstrd 3931 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086  wcel 2106  {crab 3068  wss 3887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904
This theorem is referenced by:  suppfnss  8005  clwlknon2num  28732  numclwlk1lem2  28734
  Copyright terms: Public domain W3C validator