Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  exse2 Structured version   Visualization version   GIF version

Theorem exse2 7607
 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 3141 . . . . 5 {𝑦𝐴𝑦𝑅𝑥} = {𝑦 ∣ (𝑦𝐴𝑦𝑅𝑥)}
2 vex 3482 . . . . . . . 8 𝑦 ∈ V
3 vex 3482 . . . . . . . 8 𝑥 ∈ V
42, 3breldm 5760 . . . . . . 7 (𝑦𝑅𝑥𝑦 ∈ dom 𝑅)
54adantl 485 . . . . . 6 ((𝑦𝐴𝑦𝑅𝑥) → 𝑦 ∈ dom 𝑅)
65abssi 4030 . . . . 5 {𝑦 ∣ (𝑦𝐴𝑦𝑅𝑥)} ⊆ dom 𝑅
71, 6eqsstri 3985 . . . 4 {𝑦𝐴𝑦𝑅𝑥} ⊆ dom 𝑅
8 dmexg 7598 . . . 4 (𝑅𝑉 → dom 𝑅 ∈ V)
9 ssexg 5210 . . . 4 (({𝑦𝐴𝑦𝑅𝑥} ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
107, 8, 9sylancr 590 . . 3 (𝑅𝑉 → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
1110ralrimivw 3177 . 2 (𝑅𝑉 → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
12 df-se 5498 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
1311, 12sylibr 237 1 (𝑅𝑉𝑅 Se 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2115  {cab 2802  ∀wral 3132  {crab 3136  Vcvv 3479   ⊆ wss 3918   class class class wbr 5049   Se wse 5495  dom cdm 5538 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pr 5313  ax-un 7446 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-opab 5112  df-se 5498  df-cnv 5546  df-dm 5548  df-rn 5549 This theorem is referenced by:  dfac8clem  9445
 Copyright terms: Public domain W3C validator