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Theorem mpt2fun 6403
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 2485 . . . . . 6  |-  ( ( z  =  C  /\  w  =  C )  ->  z  =  w )
21ad2ant2l 745 . . . . 5  |-  ( ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  /\  (
( x  e.  A  /\  y  e.  B
)  /\  w  =  C ) )  -> 
z  =  w )
32gen2 1620 . . . 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 2461 . . . . . 6  |-  ( z  =  w  ->  (
z  =  C  <->  w  =  C ) )
54anbi2d 703 . . . . 5  |-  ( z  =  w  ->  (
( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( (
x  e.  A  /\  y  e.  B )  /\  w  =  C
) ) )
65mo4 2338 . . . 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 209 . . 3  |-  E* z
( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )
87funoprab 6401 . 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 6301 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
119, 10eqtri 2486 . . 3  |-  F  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }
1211funeqi 5614 . 2  |-  ( Fun 
F  <->  Fun  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) } )
138, 12mpbir 209 1  |-  Fun  F
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1393    = wceq 1395    e. wcel 1819   E*wmo 2284   Fun wfun 5588   {coprab 6297    |-> cmpt2 6298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-fun 5596  df-oprab 6300  df-mpt2 6301
This theorem is referenced by:  ofexg  6543  mpt2exxg  6873  mpt2curryd  7016  imasvscafn  14954  coapm  15477  oppglsm  16789  gsum2d2lem  17128  evlslem2  18307  xkococnlem  20286  ucnima  20910  ucnprima  20911  fmucnd  20921  txomap  27998  tpr2rico  28055  elunirnmbfm  28397  aovmpt4g  32489  mpt2exxg2  33071
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