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Theorem fmpt 5687
Description: Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
fmpt.1  |-  F  =  ( x  e.  A  |->  C )
Assertion
Ref Expression
fmpt  |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
Distinct variable groups:    x, A    x, B
Allowed substitution hints:    C( x)    F( x)

Proof of Theorem fmpt
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fmpt.1 . . . 4  |-  F  =  ( x  e.  A  |->  C )
21fnmpt 5361 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  F  Fn  A )
31rnmpt 4893 . . . 4  |-  ran  F  =  { y  |  E. x  e.  A  y  =  C }
4 r19.29 2627 . . . . . . 7  |-  ( ( A. x  e.  A  C  e.  B  /\  E. x  e.  A  y  =  C )  ->  E. x  e.  A  ( C  e.  B  /\  y  =  C
) )
5 eleq1 2252 . . . . . . . . 9  |-  ( y  =  C  ->  (
y  e.  B  <->  C  e.  B ) )
65biimparc 299 . . . . . . . 8  |-  ( ( C  e.  B  /\  y  =  C )  ->  y  e.  B )
76rexlimivw 2603 . . . . . . 7  |-  ( E. x  e.  A  ( C  e.  B  /\  y  =  C )  ->  y  e.  B )
84, 7syl 14 . . . . . 6  |-  ( ( A. x  e.  A  C  e.  B  /\  E. x  e.  A  y  =  C )  -> 
y  e.  B )
98ex 115 . . . . 5  |-  ( A. x  e.  A  C  e.  B  ->  ( E. x  e.  A  y  =  C  ->  y  e.  B ) )
109abssdv 3244 . . . 4  |-  ( A. x  e.  A  C  e.  B  ->  { y  |  E. x  e.  A  y  =  C }  C_  B )
113, 10eqsstrid 3216 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  ran  F  C_  B )
12 df-f 5239 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
132, 11, 12sylanbrc 417 . 2  |-  ( A. x  e.  A  C  e.  B  ->  F : A
--> B )
14 fimacnv 5666 . . . 4  |-  ( F : A --> B  -> 
( `' F " B )  =  A )
151mptpreima 5140 . . . 4  |-  ( `' F " B )  =  { x  e.  A  |  C  e.  B }
1614, 15eqtr3di 2237 . . 3  |-  ( F : A --> B  ->  A  =  { x  e.  A  |  C  e.  B } )
17 rabid2 2667 . . 3  |-  ( A  =  { x  e.  A  |  C  e.  B }  <->  A. x  e.  A  C  e.  B )
1816, 17sylib 122 . 2  |-  ( F : A --> B  ->  A. x  e.  A  C  e.  B )
1913, 18impbii 126 1  |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   {cab 2175   A.wral 2468   E.wrex 2469   {crab 2472    C_ wss 3144    |-> cmpt 4079   `'ccnv 4643   ran crn 4645   "cima 4647    Fn wfn 5230   -->wf 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243
This theorem is referenced by:  f1ompt  5688  fmpti  5689  fvmptelcdm  5690  fmptd  5691  fmptdf  5694  rnmptss  5698  f1oresrab  5702  idref  5778  f1mpt  5793  f1stres  6185  f2ndres  6186  fmpox  6226  fmpoco  6242  iunon  6310  mptelixpg  6761  dom2lem  6799  uzf  9562  pcmptcl  12377  upxp  14249  txdis1cn  14255  cnmpt11  14260  cnmpt21  14268  fsumcncntop  14533  cncfmpt1f  14561  mulcncflem  14567  mulcncf  14568  cnmptlimc  14620  sincn  14667  coscn  14668
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