MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnopab Structured version   Unicode version

Theorem fnopab 5569
Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)
Hypotheses
Ref Expression
fnopab.1  |-  ( x  e.  A  ->  E! y ph )
fnopab.2  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }
Assertion
Ref Expression
fnopab  |-  F  Fn  A
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)    F( x, y)

Proof of Theorem fnopab
StepHypRef Expression
1 fnopab.1 . . 3  |-  ( x  e.  A  ->  E! y ph )
21rgen 2771 . 2  |-  A. x  e.  A  E! y ph
3 fnopab.2 . . 3  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }
43fnopabg 5568 . 2  |-  ( A. x  e.  A  E! y ph  <->  F  Fn  A
)
52, 4mpbi 200 1  |-  F  Fn  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   E!weu 2281   A.wral 2705   {copab 4265    Fn wfn 5449
This theorem is referenced by:  fvopab3g  5802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-fun 5456  df-fn 5457
  Copyright terms: Public domain W3C validator