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Theorem brab1 4630
 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 3176 . . 3 𝑥 ∈ V
2 breq1 4586 . . . 4 (𝑧 = 𝑦 → (𝑧𝑅𝐴𝑦𝑅𝐴))
3 breq1 4586 . . . 4 (𝑦 = 𝑥 → (𝑦𝑅𝐴𝑥𝑅𝐴))
42, 3sbcie2g 3436 . . 3 (𝑥 ∈ V → ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥𝑅𝐴))
51, 4ax-mp 5 . 2 ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥𝑅𝐴)
6 df-sbc 3403 . 2 ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
75, 6bitr3i 265 1 (𝑥𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∈ wcel 1977  {cab 2596  Vcvv 3173  [wsbc 3402   class class class wbr 4583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584 This theorem is referenced by: (None)
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