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Theorem exse2 7147
Description: Any set relation is set-like. (Contributed by Mario Carneiro, 22-Jun-2015.)
Assertion
Ref Expression
exse2 (𝑅𝑉𝑅 Se 𝐴)

Proof of Theorem exse2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rab 2950 . . . . 5 {𝑦𝐴𝑦𝑅𝑥} = {𝑦 ∣ (𝑦𝐴𝑦𝑅𝑥)}
2 vex 3234 . . . . . . . 8 𝑦 ∈ V
3 vex 3234 . . . . . . . 8 𝑥 ∈ V
42, 3breldm 5361 . . . . . . 7 (𝑦𝑅𝑥𝑦 ∈ dom 𝑅)
54adantl 481 . . . . . 6 ((𝑦𝐴𝑦𝑅𝑥) → 𝑦 ∈ dom 𝑅)
65abssi 3710 . . . . 5 {𝑦 ∣ (𝑦𝐴𝑦𝑅𝑥)} ⊆ dom 𝑅
71, 6eqsstri 3668 . . . 4 {𝑦𝐴𝑦𝑅𝑥} ⊆ dom 𝑅
8 dmexg 7139 . . . 4 (𝑅𝑉 → dom 𝑅 ∈ V)
9 ssexg 4837 . . . 4 (({𝑦𝐴𝑦𝑅𝑥} ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
107, 8, 9sylancr 696 . . 3 (𝑅𝑉 → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
1110ralrimivw 2996 . 2 (𝑅𝑉 → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
12 df-se 5103 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
1311, 12sylibr 224 1 (𝑅𝑉𝑅 Se 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  {cab 2637  wral 2941  {crab 2945  Vcvv 3231  wss 3607   class class class wbr 4685   Se wse 5100  dom cdm 5143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-se 5103  df-cnv 5151  df-dm 5153  df-rn 5154
This theorem is referenced by:  dfac8clem  8893
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