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Theorem mpt2fun 6105
Description: The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)
Hypothesis
Ref Expression
mpt2fun.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
mpt2fun  |-  Fun  F
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)    C( x, y)    F( x, y)

Proof of Theorem mpt2fun
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqtr3 2400 . . . . . 6  |-  ( ( z  =  C  /\  w  =  C )  ->  z  =  w )
21ad2ant2l 727 . . . . 5  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  /\  (
( x  e.  A  /\  y  e.  B
)  /\  w  =  C ) )  -> 
z  =  w )
32gen2 1553 . . . 4  |-  A. z A. w ( ( ( ( x  e.  A  /\  y  e.  B
)  /\  z  =  C )  /\  (
( x  e.  A  /\  y  e.  B
)  /\  w  =  C ) )  -> 
z  =  w )
4 eqeq1 2387 . . . . . 6  |-  ( z  =  w  ->  (
z  =  C  <->  w  =  C ) )
54anbi2d 685 . . . . 5  |-  ( z  =  w  ->  (
( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( (
x  e.  A  /\  y  e.  B )  /\  w  =  C
) ) )
65mo4 2265 . . . 4  |-  ( E* z ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  A. z A. w ( ( ( ( x  e.  A  /\  y  e.  B
)  /\  z  =  C )  /\  (
( x  e.  A  /\  y  e.  B
)  /\  w  =  C ) )  -> 
z  =  w ) )
73, 6mpbir 201 . . 3  |-  E* z
( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )
87funoprab 6103 . 2  |-  Fun  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
9 mpt2fun.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
10 df-mpt2 6019 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
119, 10eqtri 2401 . . 3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
1211funeqi 5408 . 2  |-  ( Fun 
F  <->  Fun  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) } )
138, 12mpbir 201 1  |-  Fun  F
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717   E*wmo 2233   Fun wfun 5382   {coprab 6015    e. cmpt2 6016
This theorem is referenced by:  ofexg  6242  mpt2exxg  6355  imasvscafn  13683  coapm  14147  oppglsm  15197  xkococnlem  17606  ucnima  18226  ucnprima  18227  fmucnd  18237  tpr2rico  24108  elunirnmbfm  24391  aovmpt4g  27728
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-sep 4265  ax-nul 4273  ax-pr 4338
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-rab 2652  df-v 2895  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-sn 3757  df-pr 3758  df-op 3760  df-br 4148  df-opab 4202  df-id 4433  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-fun 5390  df-oprab 6018  df-mpt2 6019
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