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Theorem dfse2 4748
Description: Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)
Assertion
Ref Expression
dfse2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem dfse2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-se 4116 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
2 dfrab3 3256 . . . . 5 {𝑦𝐴𝑦𝑅𝑥} = (𝐴 ∩ {𝑦𝑦𝑅𝑥})
3 vex 2613 . . . . . . 7 𝑥 ∈ V
4 iniseg 4747 . . . . . . 7 (𝑥 ∈ V → (𝑅 “ {𝑥}) = {𝑦𝑦𝑅𝑥})
53, 4ax-mp 7 . . . . . 6 (𝑅 “ {𝑥}) = {𝑦𝑦𝑅𝑥}
65ineq2i 3180 . . . . 5 (𝐴 ∩ (𝑅 “ {𝑥})) = (𝐴 ∩ {𝑦𝑦𝑅𝑥})
72, 6eqtr4i 2106 . . . 4 {𝑦𝐴𝑦𝑅𝑥} = (𝐴 ∩ (𝑅 “ {𝑥}))
87eleq1i 2148 . . 3 ({𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
98ralbii 2377 . 2 (∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
101, 9bitri 182 1 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
Colors of variables: wff set class
Syntax hints:  wb 103   = wceq 1285  wcel 1434  {cab 2069  wral 2353  {crab 2357  Vcvv 2610  cin 2981  {csn 3416   class class class wbr 3805   Se wse 4112  ccnv 4390  cima 4394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-se 4116  df-xp 4397  df-cnv 4399  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404
This theorem is referenced by:  isoselem  5510
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