MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfse2 Structured version   Visualization version   GIF version

Theorem dfse2 6104
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 5634 . 2 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V)
2 dfrab3 4310 . . . . 5 {𝑦𝐴𝑦𝑅𝑥} = (𝐴 ∩ {𝑦𝑦𝑅𝑥})
3 iniseg 6101 . . . . . . 7 (𝑥 ∈ V → (𝑅 “ {𝑥}) = {𝑦𝑦𝑅𝑥})
43elv 3477 . . . . . 6 (𝑅 “ {𝑥}) = {𝑦𝑦𝑅𝑥}
54ineq2i 4209 . . . . 5 (𝐴 ∩ (𝑅 “ {𝑥})) = (𝐴 ∩ {𝑦𝑦𝑅𝑥})
62, 5eqtr4i 2759 . . . 4 {𝑦𝐴𝑦𝑅𝑥} = (𝐴 ∩ (𝑅 “ {𝑥}))
76eleq1i 2820 . . 3 ({𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
87ralbii 3090 . 2 (∀𝑥𝐴 {𝑦𝐴𝑦𝑅𝑥} ∈ V ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
91, 8bitri 275 1 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 (𝐴 ∩ (𝑅 “ {𝑥})) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  wcel 2099  {cab 2705  wral 3058  {crab 3429  Vcvv 3471  cin 3946  {csn 4629   class class class wbr 5148   Se wse 5631  ccnv 5677  cima 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-se 5634  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691
This theorem is referenced by:  dfse3  6342  isoselem  7349  fnse  8138
  Copyright terms: Public domain W3C validator