MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funopab4 Unicode version

Theorem funopab4 5455
Description: A class of ordered pairs of values in the form used by df-mpt 4236 is a function. (Contributed by NM, 17-Feb-2013.)
Assertion
Ref Expression
funopab4  |-  Fun  { <. x ,  y >.  |  ( ph  /\  y  =  A ) }
Distinct variable groups:    x, y    y, A
Allowed substitution hints:    ph( x, y)    A( x)

Proof of Theorem funopab4
StepHypRef Expression
1 simpr 448 . . 3  |-  ( (
ph  /\  y  =  A )  ->  y  =  A )
21ssopab2i 4450 . 2  |-  { <. x ,  y >.  |  (
ph  /\  y  =  A ) }  C_  {
<. x ,  y >.  |  y  =  A }
3 funopabeq 5454 . 2  |-  Fun  { <. x ,  y >.  |  y  =  A }
4 funss 5439 . 2  |-  ( {
<. x ,  y >.  |  ( ph  /\  y  =  A ) }  C_  { <. x ,  y >.  |  y  =  A }  ->  ( Fun  { <. x ,  y >.  |  y  =  A }  ->  Fun 
{ <. x ,  y
>.  |  ( ph  /\  y  =  A ) } ) )
52, 3, 4mp2 9 1  |-  Fun  { <. x ,  y >.  |  ( ph  /\  y  =  A ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    C_ wss 3288   {copab 4233   Fun wfun 5415
This theorem is referenced by:  funmpt  5456  hartogslem1  7475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-fun 5423
  Copyright terms: Public domain W3C validator