Theorem exse2 7918

Description: Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
exse2	⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴)

Proof of Theorem exse2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 3421	. . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)}
2		vex 3468	. . . . . . . 8 ⊢ 𝑦 ∈ V
3		vex 3468	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	2, 3	breldm 5893	. . . . . . 7 ⊢ (𝑦𝑅𝑥 → 𝑦 ∈ dom 𝑅)
5	4	adantl 481	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) → 𝑦 ∈ dom 𝑅)
6	5	abssi 4050	. . . . 5 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)} ⊆ dom 𝑅
7	1, 6	eqsstri 4010	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ dom 𝑅
8		dmexg 7902	. . . 4 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
9		ssexg 5298	. . . 4 ⊢ (({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
10	7, 8, 9	sylancr 587	. . 3 ⊢ (𝑅 ∈ 𝑉 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
11	10	ralrimivw 3137	. 2 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
12		df-se 5612	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
13	11, 12	sylibr 234	1 ⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  {cab 2714  ∀wral 3052  {crab 3420  Vcvv 3464   ⊆ wss 3931   class class class wbr 5124   Se wse 5609  dom cdm 5659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-se 5612  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is referenced by:  dfac8clem  10051