Theorem exse2 7604

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

Assertion

Ref	Expression
exse2	⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴)

Proof of Theorem exse2

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

Step	Hyp	Ref	Expression
1		df-rab 3115	. . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)}
2		vex 3444	. . . . . . . 8 ⊢ 𝑦 ∈ V
3		vex 3444	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	2, 3	breldm 5741	. . . . . . 7 ⊢ (𝑦𝑅𝑥 → 𝑦 ∈ dom 𝑅)
5	4	adantl 485	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) → 𝑦 ∈ dom 𝑅)
6	5	abssi 3997	. . . . 5 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)} ⊆ dom 𝑅
7	1, 6	eqsstri 3949	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ dom 𝑅
8		dmexg 7594	. . . 4 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
9		ssexg 5191	. . . 4 ⊢ (({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
10	7, 8, 9	sylancr 590	. . 3 ⊢ (𝑅 ∈ 𝑉 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
11	10	ralrimivw 3150	. 2 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
12		df-se 5479	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
13	11, 12	sylibr 237	1 ⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  {cab 2776  ∀wral 3106  {crab 3110  Vcvv 3441   ⊆ wss 3881   class class class wbr 5030   Se wse 5476  dom cdm 5519

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-se 5479  df-cnv 5527  df-dm 5529  df-rn 5530

This theorem is referenced by:  dfac8clem  9443