Theorem exse 5579

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 5276	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
2	1	ralrimivw 3125	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
3		df-se 5573	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
4	2, 3	sylibr 234	1 ⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2109  ∀wral 3044  {crab 3394  Vcvv 3436   class class class wbr 5092   Se wse 5570

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 2701  ax-sep 5235

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3395  df-v 3438  df-in 3910  df-ss 3920  df-pw 4553  df-se 5573

This theorem is referenced by:  wemoiso  7908  wemoiso2  7909  oiiso  9429  hartogslem1  9434  oemapwe  9590  cantnffval2  9591  om2uzoi  13862  uzsinds  13894  bpolylem  15955  om2noseqoi  28204  numiunnum  36464