Theorem brab1 4080

Description: Relationship between a binary relation and a class abstraction. (Contributed by Andrew Salmon, 8-Jul-2011.)

Assertion

Ref	Expression
brab1	⊢ (𝑥𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴})

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.

Step	Hyp	Ref	Expression
1		vex 2766	. . 3 ⊢ 𝑥 ∈ V
2		breq1 4036	. . . 4 ⊢ (𝑧 = 𝑦 → (𝑧𝑅𝐴 ↔ 𝑦𝑅𝐴))
3		breq1 4036	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝐴 ↔ 𝑥𝑅𝐴))
4	2, 3	sbcie2g 3023	. . 3 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥𝑅𝐴))
5	1, 4	ax-mp 5	. 2 ⊢ ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥𝑅𝐴)
6		df-sbc 2990	. 2 ⊢ ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴})
7	5, 6	bitr3i 186	1 ⊢ (𝑥𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴})

Syntax hints:   ↔ wb 105   ∈ wcel 2167  {cab 2182  Vcvv 2763  [wsbc 2989   class class class wbr 4033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034

This theorem is referenced by: (None)