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Theorem brab1 5081
 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 breq1 5036 . . . 4 (𝑧 = 𝑦 → (𝑧𝑅𝐴𝑦𝑅𝐴))
2 breq1 5036 . . . 4 (𝑦 = 𝑥 → (𝑦𝑅𝐴𝑥𝑅𝐴))
31, 2sbcie2g 3762 . . 3 (𝑥 ∈ V → ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥𝑅𝐴))
43elv 3449 . 2 ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥𝑅𝐴)
5 df-sbc 3724 . 2 ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
64, 5bitr3i 280 1 (𝑥𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∈ wcel 2112  {cab 2779  Vcvv 3444  [wsbc 3723   class class class wbr 5033 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446  df-sbc 3724  df-un 3889  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034 This theorem is referenced by: (None)
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