Theorem exse 5638

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 5330	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
2	1	ralrimivw 3150	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
3		df-se 5631	. 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 3061  {crab 3432  Vcvv 3474   class class class wbr 5147   Se wse 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-in 3954  df-ss 3964  df-se 5631

This theorem is referenced by:  wemoiso  7956  wemoiso2  7957  oiiso  9528  hartogslem1  9533  oemapwe  9685  cantnffval2  9686  om2uzoi  13916  uzsinds  13948  bpolylem  15988