Theorem exse2 7915

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 3413	. . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)}
2		vex 3454	. . . . . . . 8 ⊢ 𝑦 ∈ V
3		vex 3454	. . . . . . . 8 ⊢ 𝑥 ∈ V
4	2, 3	breldm 5892	. . . . . . 7 ⊢ (𝑦𝑅𝑥 → 𝑦 ∈ dom 𝑅)
5	4	adantl 487	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) → 𝑦 ∈ dom 𝑅)
6	5	abssi 4016	. . . . 5 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥)} ⊆ dom 𝑅
7	1, 6	eqsstri 3977	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ dom 𝑅
8		dmexg 7899	. . . 4 ⊢ (𝑅 ∈ 𝑉 → dom 𝑅 ∈ V)
9		ssexg 5284	. . . 4 ⊢ (({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
10	7, 8, 9	sylancr 599	. . 3 ⊢ (𝑅 ∈ 𝑉 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
11	10	ralrimivw 3158	. 2 ⊢ (𝑅 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
12		df-se 5609	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
13	11, 12	sylibr 237	1 ⊢ (𝑅 ∈ 𝑉 → 𝑅 Se 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  {cab 2738  ∀wral 3076  {crab 3412  Vcvv 3450   ⊆ wss 3899   class class class wbr 5103   Se wse 5606  dom cdm 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-se 5609  df-cnv 5663  df-dm 5665  df-rn 5666

This theorem is used by:  dfac8clem  10036