Theorem brab1 5199

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 5154	. . . 4 ⊢ (𝑧 = 𝑦 → (𝑧𝑅𝐴 ↔ 𝑦𝑅𝐴))
2		breq1 5154	. . . 4 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝐴 ↔ 𝑥𝑅𝐴))
3	1, 2	sbcie2g 3837	. . 3 ⊢ (𝑥 ∈ V → ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥𝑅𝐴))
4	3	elv 3486	. 2 ⊢ ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥𝑅𝐴)
5		df-sbc 3795	. 2 ⊢ ([𝑥 / 𝑧]𝑧𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴})
6	4, 5	bitr3i 277	1 ⊢ (𝑥𝑅𝐴 ↔ 𝑥 ∈ {𝑧 ∣ 𝑧𝑅𝐴})

Syntax hints:   ↔ wb 206   ∈ wcel 2108  {cab 2714  Vcvv 3481  [wsbc 3794   class class class wbr 5151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3483  df-sbc 3795  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-br 5152

This theorem is referenced by: (None)