MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brab1 Structured version   Visualization version   GIF version

Theorem brab1 4670
Description: Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)
Assertion
Ref Expression
brab1 (𝑥𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
Distinct variable groups:   𝑧,𝐴   𝑧,𝑅
Allowed substitution hints:   𝐴(𝑥)   𝑅(𝑥)

Proof of Theorem brab1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 3193 . . 3 𝑥 ∈ V
2 breq1 4626 . . . 4 (𝑧 = 𝑦 → (𝑧𝑅𝐴𝑦𝑅𝐴))
3 breq1 4626 . . . 4 (𝑦 = 𝑥 → (𝑦𝑅𝐴𝑥𝑅𝐴))
42, 3sbcie2g 3456 . . 3 (𝑥 ∈ V → ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥𝑅𝐴))
51, 4ax-mp 5 . 2 ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥𝑅𝐴)
6 df-sbc 3423 . 2 ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
75, 6bitr3i 266 1 (𝑥𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 1987  {cab 2607  Vcvv 3190  [wsbc 3422   class class class wbr 4623
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-3an 1038  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-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator