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Theorem fmpt 5449
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 5140 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  F  Fn  A )
31rnmpt 4683 . . . 4  |-  ran  F  =  { y  |  E. x  e.  A  y  =  C }
4 r19.29 2506 . . . . . . 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 2150 . . . . . . . . 9  |-  ( y  =  C  ->  (
y  e.  B  <->  C  e.  B ) )
65biimparc 293 . . . . . . . 8  |-  ( ( C  e.  B  /\  y  =  C )  ->  y  e.  B )
76rexlimivw 2485 . . . . . . 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 113 . . . . 5  |-  ( A. x  e.  A  C  e.  B  ->  ( E. x  e.  A  y  =  C  ->  y  e.  B ) )
109abssdv 3095 . . . 4  |-  ( A. x  e.  A  C  e.  B  ->  { y  |  E. x  e.  A  y  =  C }  C_  B )
113, 10syl5eqss 3070 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  ran  F  C_  B )
12 df-f 5019 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
132, 11, 12sylanbrc 408 . 2  |-  ( A. x  e.  A  C  e.  B  ->  F : A
--> B )
141mptpreima 4924 . . . 4  |-  ( `' F " B )  =  { x  e.  A  |  C  e.  B }
15 fimacnv 5428 . . . 4  |-  ( F : A --> B  -> 
( `' F " B )  =  A )
1614, 15syl5reqr 2135 . . 3  |-  ( F : A --> B  ->  A  =  { x  e.  A  |  C  e.  B } )
17 rabid2 2543 . . 3  |-  ( A  =  { x  e.  A  |  C  e.  B }  <->  A. x  e.  A  C  e.  B )
1816, 17sylib 120 . 2  |-  ( F : A --> B  ->  A. x  e.  A  C  e.  B )
1913, 18impbii 124 1  |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363    C_ wss 2999    |-> cmpt 3899   `'ccnv 4437   ran crn 4439   "cima 4441    Fn wfn 5010   -->wf 5011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023
This theorem is referenced by:  f1ompt  5450  fmpti  5451  fmptd  5452  fmptdf  5455  rnmptss  5459  f1oresrab  5463  idref  5536  f1mpt  5550  f1stres  5930  f2ndres  5931  fmpt2x  5970  fmpt2co  5981  iunon  6049  dom2lem  6487  uzf  9020
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