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Theorem exse2 7059
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 2916 . . . . 5 {𝑦𝐴𝑦𝑅𝑥} = {𝑦 ∣ (𝑦𝐴𝑦𝑅𝑥)}
2 vex 3192 . . . . . . . 8 𝑦 ∈ V
3 vex 3192 . . . . . . . 8 𝑥 ∈ V
42, 3breldm 5294 . . . . . . 7 (𝑦𝑅𝑥𝑦 ∈ dom 𝑅)
54adantl 482 . . . . . 6 ((𝑦𝐴𝑦𝑅𝑥) → 𝑦 ∈ dom 𝑅)
65abssi 3661 . . . . 5 {𝑦 ∣ (𝑦𝐴𝑦𝑅𝑥)} ⊆ dom 𝑅
71, 6eqsstri 3619 . . . 4 {𝑦𝐴𝑦𝑅𝑥} ⊆ dom 𝑅
8 dmexg 7051 . . . 4 (𝑅𝑉 → dom 𝑅 ∈ V)
9 ssexg 4769 . . . 4 (({𝑦𝐴𝑦𝑅𝑥} ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
107, 8, 9sylancr 694 . . 3 (𝑅𝑉 → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
1110ralrimivw 2962 . 2 (𝑅𝑉 → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
12 df-se 5039 . 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 384  wcel 1987  {cab 2607  wral 2907  {crab 2911  Vcvv 3189  wss 3559   class class class wbr 4618   Se wse 5036  dom cdm 5079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-se 5039  df-cnv 5087  df-dm 5089  df-rn 5090
This theorem is referenced by:  dfac8clem  8807
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