Theorem exse 5002

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 4739	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
2	1	ralrimivw 2950	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
3		df-se 4998	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
4	2, 3	sylibr 223	1 ⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴)

Syntax hints:   → wi 4   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   class class class wbr 4583   Se wse 4995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-in 3547  df-ss 3554  df-se 4998

This theorem is referenced by:  wemoiso  7044  wemoiso2  7045  oiiso  8325  hartogslem1  8330  oemapwe  8474  cantnffval2  8475  om2uzoi  12616  uzsinds  12648  bpolylem  14618