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Theorem exse2 7925
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 3419 . . . . 5 {𝑦𝐴𝑦𝑅𝑥} = {𝑦 ∣ (𝑦𝐴𝑦𝑅𝑥)}
2 vex 3465 . . . . . . . 8 𝑦 ∈ V
3 vex 3465 . . . . . . . 8 𝑥 ∈ V
42, 3breldm 5911 . . . . . . 7 (𝑦𝑅𝑥𝑦 ∈ dom 𝑅)
54adantl 480 . . . . . 6 ((𝑦𝐴𝑦𝑅𝑥) → 𝑦 ∈ dom 𝑅)
65abssi 4063 . . . . 5 {𝑦 ∣ (𝑦𝐴𝑦𝑅𝑥)} ⊆ dom 𝑅
71, 6eqsstri 4011 . . . 4 {𝑦𝐴𝑦𝑅𝑥} ⊆ dom 𝑅
8 dmexg 7909 . . . 4 (𝑅𝑉 → dom 𝑅 ∈ V)
9 ssexg 5324 . . . 4 (({𝑦𝐴𝑦𝑅𝑥} ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
107, 8, 9sylancr 585 . . 3 (𝑅𝑉 → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
1110ralrimivw 3139 . 2 (𝑅𝑉 → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
12 df-se 5634 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
1311, 12sylibr 233 1 (𝑅𝑉𝑅 Se 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  {cab 2702  wral 3050  {crab 3418  Vcvv 3461  wss 3944   class class class wbr 5149   Se wse 5631  dom cdm 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-se 5634  df-cnv 5686  df-dm 5688  df-rn 5689
This theorem is referenced by:  dfac8clem  10062
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