Theorem exse 5368

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 5087	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
2	1	ralrimivw 3128	. 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
3		df-se 5364	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
4	2, 3	sylibr 226	1 ⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2051  ∀wral 3083  {crab 3087  Vcvv 3410   class class class wbr 4926   Se wse 5361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745  ax-sep 5057

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rab 3092  df-v 3412  df-in 3831  df-ss 3838  df-se 5364

This theorem is referenced by:  wemoiso  7485  wemoiso2  7486  oiiso  8795  hartogslem1  8800  oemapwe  8950  cantnffval2  8951  om2uzoi  13137  uzsinds  13169  bpolylem  15261