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

Theorem funopab4 5623
Description: A class of ordered pairs of values in the form used by df-mpt 4507 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 461 . . 3  |-  ( (
ph  /\  y  =  A )  ->  y  =  A )
21ssopab2i 4775 . 2  |-  { <. x ,  y >.  |  (
ph  /\  y  =  A ) }  C_  {
<. x ,  y >.  |  y  =  A }
3 funopabeq 5622 . 2  |-  Fun  { <. x ,  y >.  |  y  =  A }
4 funss 5606 . 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 setvar class
Syntax hints:    /\ wa 369    = wceq 1379    C_ wss 3476   {copab 4504   Fun wfun 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-fun 5590
This theorem is referenced by:  funmpt  5624  hartogslem1  7968
  Copyright terms: Public domain W3C validator