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Theorem brab1 4034
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 2733 . . 3 𝑥 ∈ V
2 breq1 3990 . . . 4 (𝑧 = 𝑦 → (𝑧𝑅𝐴𝑦𝑅𝐴))
3 breq1 3990 . . . 4 (𝑦 = 𝑥 → (𝑦𝑅𝐴𝑥𝑅𝐴))
42, 3sbcie2g 2988 . . 3 (𝑥 ∈ V → ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥𝑅𝐴))
51, 4ax-mp 5 . 2 ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥𝑅𝐴)
6 df-sbc 2956 . 2 ([𝑥 / 𝑧]𝑧𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
75, 6bitr3i 185 1 (𝑥𝑅𝐴𝑥 ∈ {𝑧𝑧𝑅𝐴})
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 2141  {cab 2156  Vcvv 2730  [wsbc 2955   class class class wbr 3987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988
This theorem is referenced by: (None)
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