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

Theorem fmpt 5488
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 5174 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  F  Fn  A )
31rnmpt 4715 . . . 4  |-  ran  F  =  { y  |  E. x  e.  A  y  =  C }
4 r19.29 2520 . . . . . . 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 2157 . . . . . . . . 9  |-  ( y  =  C  ->  (
y  e.  B  <->  C  e.  B ) )
65biimparc 294 . . . . . . . 8  |-  ( ( C  e.  B  /\  y  =  C )  ->  y  e.  B )
76rexlimivw 2498 . . . . . . 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 3110 . . . 4  |-  ( A. x  e.  A  C  e.  B  ->  { y  |  E. x  e.  A  y  =  C }  C_  B )
113, 10syl5eqss 3085 . . 3  |-  ( A. x  e.  A  C  e.  B  ->  ran  F  C_  B )
12 df-f 5053 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
132, 11, 12sylanbrc 409 . 2  |-  ( A. x  e.  A  C  e.  B  ->  F : A
--> B )
141mptpreima 4958 . . . 4  |-  ( `' F " B )  =  { x  e.  A  |  C  e.  B }
15 fimacnv 5467 . . . 4  |-  ( F : A --> B  -> 
( `' F " B )  =  A )
1614, 15syl5reqr 2142 . . 3  |-  ( F : A --> B  ->  A  =  { x  e.  A  |  C  e.  B } )
17 rabid2 2557 . . 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 1296    e. wcel 1445   {cab 2081   A.wral 2370   E.wrex 2371   {crab 2374    C_ wss 3013    |-> cmpt 3921   `'ccnv 4466   ran crn 4468   "cima 4470    Fn wfn 5044   -->wf 5045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fv 5057
This theorem is referenced by:  f1ompt  5489  fmpti  5490  fmptd  5491  fmptdf  5494  rnmptss  5498  f1oresrab  5502  idref  5574  f1mpt  5588  f1stres  5968  f2ndres  5969  fmpt2x  6008  fmpt2co  6019  iunon  6087  mptelixpg  6531  dom2lem  6569  uzf  9121  cnmpt11  12105  cncfmpt1f  12348  mulcncflem  12353  mulcncf  12354
  Copyright terms: Public domain W3C validator