ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fmpt Unicode version

Theorem fmpt 5347
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 5053 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  F  Fn  A )
31rnmpt 4610 . . . 4  |-  ran  F  =  { y  |  E. x  e.  A  y  =  C }
4 r19.29 2467 . . . . . . 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 2116 . . . . . . . . 9  |-  ( y  =  C  ->  (
y  e.  B  <->  C  e.  B ) )
65biimparc 287 . . . . . . . 8  |-  ( ( C  e.  B  /\  y  =  C )  ->  y  e.  B )
76rexlimivw 2446 . . . . . . 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 112 . . . . 5  |-  ( A. x  e.  A  C  e.  B  ->  ( E. x  e.  A  y  =  C  ->  y  e.  B ) )
109abssdv 3042 . . . 4  |-  ( A. x  e.  A  C  e.  B  ->  { y  |  E. x  e.  A  y  =  C }  C_  B )
113, 10syl5eqss 3017 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  ran  F  C_  B )
12 df-f 4934 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
132, 11, 12sylanbrc 402 . 2  |-  ( A. x  e.  A  C  e.  B  ->  F : A
--> B )
141mptpreima 4842 . . . 4  |-  ( `' F " B )  =  { x  e.  A  |  C  e.  B }
15 fimacnv 5324 . . . 4  |-  ( F : A --> B  -> 
( `' F " B )  =  A )
1614, 15syl5reqr 2103 . . 3  |-  ( F : A --> B  ->  A  =  { x  e.  A  |  C  e.  B } )
17 rabid2 2503 . . 3  |-  ( A  =  { x  e.  A  |  C  e.  B }  <->  A. x  e.  A  C  e.  B )
1816, 17sylib 131 . 2  |-  ( F : A --> B  ->  A. x  e.  A  C  e.  B )
1913, 18impbii 121 1  |-  ( A. x  e.  A  C  e.  B  <->  F : A --> B )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   E.wrex 2324   {crab 2327    C_ wss 2945    |-> cmpt 3846   `'ccnv 4372   ran crn 4374   "cima 4376    Fn wfn 4925   -->wf 4926
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fv 4938
This theorem is referenced by:  f1ompt  5348  fmpti  5349  fmptd  5350  rnmptss  5354  f1oresrab  5357  idref  5424  f1mpt  5438  f1stres  5814  f2ndres  5815  fmpt2x  5854  fmpt2co  5865  iunon  5930  dom2lem  6283  uzf  8572
  Copyright terms: Public domain W3C validator