Theorem exse2 7764

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 3073	. . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)}
2		vex 3436	. . . . . . . 8 ⊢ 𝑦 ∈ V
3		vex 3436	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	2, 3	breldm 5817	. . . . . . 7 ⊢ (𝑦𝑅𝑥 → 𝑦 ∈ dom 𝑅)
5	4	adantl 482	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) → 𝑦 ∈ dom 𝑅)
6	5	abssi 4003	. . . . 5 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)} ⊆ dom 𝑅
7	1, 6	eqsstri 3955	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ dom 𝑅
8		dmexg 7750	. . . 4 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
9		ssexg 5247	. . . 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 3104	. 2 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
12		df-se 5545	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
13	11, 12	sylibr 233	1 ⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  {cab 2715  ∀wral 3064  {crab 3068  Vcvv 3432   ⊆ wss 3887   class class class wbr 5074   Se wse 5542  dom cdm 5589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-se 5545  df-cnv 5597  df-dm 5599  df-rn 5600

This theorem is referenced by:  dfac8clem  9788