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Theorem funopab4 5305
Description: A class of ordered pairs of values in the form used by df-mpt 4095 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 447 . . 3  |-  ( (
ph  /\  y  =  A )  ->  y  =  A )
21ssopab2i 4308 . 2  |-  { <. x ,  y >.  |  (
ph  /\  y  =  A ) }  C_  {
<. x ,  y >.  |  y  =  A }
3 funopabeq 5304 . 2  |-  Fun  { <. x ,  y >.  |  y  =  A }
4 funss 5289 . 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 17 1  |-  Fun  { <. x ,  y >.  |  ( ph  /\  y  =  A ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    C_ wss 3165   {copab 4092   Fun wfun 5265
This theorem is referenced by:  funmpt  5306  hartogslem1  7273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-fun 5273
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