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Theorem funopab4 5523
Description: A class of ordered pairs of values in the form used by df-mpt 4299 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 449 . . 3  |-  ( (
ph  /\  y  =  A )  ->  y  =  A )
21ssopab2i 4517 . 2  |-  { <. x ,  y >.  |  (
ph  /\  y  =  A ) }  C_  {
<. x ,  y >.  |  y  =  A }
3 funopabeq 5522 . 2  |-  Fun  { <. x ,  y >.  |  y  =  A }
4 funss 5507 . 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 360    = wceq 1654    C_ wss 3309   {copab 4296   Fun wfun 5483
This theorem is referenced by:  funmpt  5524  hartogslem1  7547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-br 4244  df-opab 4298  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-fun 5491
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