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

Theorem ssrab 3661
Description: Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
ssrab (𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssrab
StepHypRef Expression
1 df-rab 2916 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
21sseq2i 3611 . 2 (𝐵 ⊆ {𝑥𝐴𝜑} ↔ 𝐵 ⊆ {𝑥 ∣ (𝑥𝐴𝜑)})
3 ssab 3653 . 2 (𝐵 ⊆ {𝑥 ∣ (𝑥𝐴𝜑)} ↔ ∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)))
4 dfss3 3574 . . . 4 (𝐵𝐴 ↔ ∀𝑥𝐵 𝑥𝐴)
54anbi1i 730 . . 3 ((𝐵𝐴 ∧ ∀𝑥𝐵 𝜑) ↔ (∀𝑥𝐵 𝑥𝐴 ∧ ∀𝑥𝐵 𝜑))
6 r19.26 3057 . . 3 (∀𝑥𝐵 (𝑥𝐴𝜑) ↔ (∀𝑥𝐵 𝑥𝐴 ∧ ∀𝑥𝐵 𝜑))
7 df-ral 2912 . . 3 (∀𝑥𝐵 (𝑥𝐴𝜑) ↔ ∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)))
85, 6, 73bitr2ri 289 . 2 (∀𝑥(𝑥𝐵 → (𝑥𝐴𝜑)) ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
92, 3, 83bitri 286 1 (𝐵 ⊆ {𝑥𝐴𝜑} ↔ (𝐵𝐴 ∧ ∀𝑥𝐵 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wcel 1987  {cab 2607  wral 2907  {crab 2911  wss 3556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-in 3563  df-ss 3570
This theorem is referenced by:  ssrabdv  3662  omssnlim  7029  ordtypelem2  8371  ordtypelem10  8379  card2inf  8407  r0weon  8782  ramtlecl  15631  sscntz  17683  ppttop  20724  epttop  20726  cmpcov2  21106  tgcmp  21117  xkoinjcn  21403  fbssfi  21554  filssufilg  21628  uffixfr  21640  tmdgsum2  21813  symgtgp  21818  ghmcnp  21831  blcls  22224  clsocv  22962  lhop1lem  23687  ressatans  24568  axcontlem3  25753  axcontlem4  25754  ldgenpisyslem3  30021  ldgenpisys  30022  imambfm  30117  connpconn  30946  cvmlift2lem11  31024  cvmlift2lem12  31025  bj-rabtr  32594  bj-rabtrALTALT  32596  hbtlem6  37201
  Copyright terms: Public domain W3C validator