Theorem brab1 5158

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		breq1 5113	. . . 4 ⊢ (𝑧 = 𝑦 → (𝑧𝑅𝐴 ↔ 𝑦𝑅𝐴))
2		breq1 5113	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝐴 ↔ 𝑥𝑅𝐴))
3	1, 2	sbcie2g 3784	. . 3 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥𝑅𝐴))
4	3	elv 3452	. 2 ⊢ ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥𝑅𝐴)
5		df-sbc 3743	. 2 ⊢ ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴})
6	4, 5	bitr3i 276	1 ⊢ (𝑥𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴})

Syntax hints:   ↔ wb 205   ∈ wcel 2106  {cab 2708  Vcvv 3446  [wsbc 3742   class class class wbr 5110

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3406  df-v 3448  df-sbc 3743  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111

This theorem is referenced by: (None)