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Theorem funopab4 5627
Description: A class of ordered pairs of values in the form used by df-mpt 4477 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 462 . . 3  |-  ( (
ph  /\  y  =  A )  ->  y  =  A )
21ssopab2i 4740 . 2  |-  { <. x ,  y >.  |  (
ph  /\  y  =  A ) }  C_  {
<. x ,  y >.  |  y  =  A }
3 funopabeq 5626 . 2  |-  Fun  { <. x ,  y >.  |  y  =  A }
4 funss 5610 . 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 370    = wceq 1437    C_ wss 3433   {copab 4474   Fun wfun 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-fun 5594
This theorem is referenced by:  funmpt  5628  hartogslem1  8048
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