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Theorem fmpt 5538
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 5219 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  F  Fn  A )
31rnmpt 4757 . . . 4  |-  ran  F  =  { y  |  E. x  e.  A  y  =  C }
4 r19.29 2546 . . . . . . 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 2180 . . . . . . . . 9  |-  ( y  =  C  ->  (
y  e.  B  <->  C  e.  B ) )
65biimparc 297 . . . . . . . 8  |-  ( ( C  e.  B  /\  y  =  C )  ->  y  e.  B )
76rexlimivw 2522 . . . . . . 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 114 . . . . 5  |-  ( A. x  e.  A  C  e.  B  ->  ( E. x  e.  A  y  =  C  ->  y  e.  B ) )
109abssdv 3141 . . . 4  |-  ( A. x  e.  A  C  e.  B  ->  { y  |  E. x  e.  A  y  =  C }  C_  B )
113, 10eqsstrid 3113 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  ran  F  C_  B )
12 df-f 5097 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
132, 11, 12sylanbrc 413 . 2  |-  ( A. x  e.  A  C  e.  B  ->  F : A
--> B )
141mptpreima 5002 . . . 4  |-  ( `' F " B )  =  { x  e.  A  |  C  e.  B }
15 fimacnv 5517 . . . 4  |-  ( F : A --> B  -> 
( `' F " B )  =  A )
1614, 15syl5reqr 2165 . . 3  |-  ( F : A --> B  ->  A  =  { x  e.  A  |  C  e.  B } )
17 rabid2 2584 . . 3  |-  ( A  =  { x  e.  A  |  C  e.  B }  <->  A. x  e.  A  C  e.  B )
1816, 17sylib 121 . 2  |-  ( F : A --> B  ->  A. x  e.  A  C  e.  B )
1913, 18impbii 125 1  |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   {cab 2103   A.wral 2393   E.wrex 2394   {crab 2397    C_ wss 3041    |-> cmpt 3959   `'ccnv 4508   ran crn 4510   "cima 4512    Fn wfn 5088   -->wf 5089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101
This theorem is referenced by:  f1ompt  5539  fmpti  5540  fvmptelrn  5541  fmptd  5542  fmptdf  5545  rnmptss  5549  f1oresrab  5553  idref  5626  f1mpt  5640  f1stres  6025  f2ndres  6026  fmpox  6066  fmpoco  6081  iunon  6149  mptelixpg  6596  dom2lem  6634  uzf  9285  upxp  12352  txdis1cn  12358  cnmpt11  12363  cnmpt21  12371  fsumcncntop  12636  cncfmpt1f  12664  mulcncflem  12670  mulcncf  12671  cnmptlimc  12723  sincn  12769  coscn  12770
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