Theorem exse 5552

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

Assertion

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

Proof of Theorem exse

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

Step	Hyp	Ref	Expression
1		rabexg 5255	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
2	1	ralrimivw 3104	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
3		df-se 5545	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
4	2, 3	sylibr 233	1 ⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2106  ∀wral 3064  {crab 3068  Vcvv 3432   class class class wbr 5074   Se wse 5542

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

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  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-in 3894  df-ss 3904  df-se 5545

This theorem is referenced by:  wemoiso  7816  wemoiso2  7817  oiiso  9296  hartogslem1  9301  oemapwe  9452  cantnffval2  9453  om2uzoi  13675  uzsinds  13707  bpolylem  15758