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Theorem exse2 4726
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 2332 . . . . 5 {𝑦𝐴𝑦𝑅𝑥} = {𝑦 ∣ (𝑦𝐴𝑦𝑅𝑥)}
2 vex 2577 . . . . . . . 8 𝑦 ∈ V
3 vex 2577 . . . . . . . 8 𝑥 ∈ V
42, 3breldm 4566 . . . . . . 7 (𝑦𝑅𝑥𝑦 ∈ dom 𝑅)
54adantl 266 . . . . . 6 ((𝑦𝐴𝑦𝑅𝑥) → 𝑦 ∈ dom 𝑅)
65abssi 3042 . . . . 5 {𝑦 ∣ (𝑦𝐴𝑦𝑅𝑥)} ⊆ dom 𝑅
71, 6eqsstri 3002 . . . 4 {𝑦𝐴𝑦𝑅𝑥} ⊆ dom 𝑅
8 dmexg 4623 . . . 4 (𝑅𝑉 → dom 𝑅 ∈ V)
9 ssexg 3923 . . . 4 (({𝑦𝐴𝑦𝑅𝑥} ⊆ dom 𝑅 ∧ dom 𝑅 ∈ V) → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
107, 8, 9sylancr 399 . . 3 (𝑅𝑉 → {𝑦𝐴𝑦𝑅𝑥} ∈ V)
1110ralrimivw 2410 . 2 (𝑅𝑉 → ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
12 df-se 4097 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
1311, 12sylibr 141 1 (𝑅𝑉𝑅 Se 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  {cab 2042  wral 2323  {crab 2327  Vcvv 2574  wss 2944   class class class wbr 3791   Se wse 4093  dom cdm 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-se 4097  df-cnv 4380  df-dm 4382  df-rn 4383
This theorem is referenced by: (None)
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