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Theorem qliftf 6222
Description: The domain and range of the function  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
qlift.1  |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )
qlift.2  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Y )
qlift.3  |-  ( ph  ->  R  Er  X )
qlift.4  |-  ( ph  ->  X  e.  _V )
Assertion
Ref Expression
qliftf  |-  ( ph  ->  ( Fun  F  <->  F :
( X /. R
) --> Y ) )
Distinct variable groups:    ph, x    x, R    x, X    x, Y
Allowed substitution hints:    A( x)    F( x)

Proof of Theorem qliftf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 qlift.1 . . 3  |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )
2 qlift.2 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Y )
3 qlift.3 . . . 4  |-  ( ph  ->  R  Er  X )
4 qlift.4 . . . 4  |-  ( ph  ->  X  e.  _V )
51, 2, 3, 4qliftlem 6215 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
61, 5, 2fliftf 5467 . 2  |-  ( ph  ->  ( Fun  F  <->  F : ran  ( x  e.  X  |->  [ x ] R
) --> Y ) )
7 df-qs 6143 . . . . 5  |-  ( X /. R )  =  { y  |  E. x  e.  X  y  =  [ x ] R }
8 eqid 2056 . . . . . 6  |-  ( x  e.  X  |->  [ x ] R )  =  ( x  e.  X  |->  [ x ] R )
98rnmpt 4610 . . . . 5  |-  ran  (
x  e.  X  |->  [ x ] R )  =  { y  |  E. x  e.  X  y  =  [ x ] R }
107, 9eqtr4i 2079 . . . 4  |-  ( X /. R )  =  ran  ( x  e.  X  |->  [ x ] R )
1110a1i 9 . . 3  |-  ( ph  ->  ( X /. R
)  =  ran  (
x  e.  X  |->  [ x ] R ) )
1211feq2d 5063 . 2  |-  ( ph  ->  ( F : ( X /. R ) --> Y  <->  F : ran  (
x  e.  X  |->  [ x ] R ) --> Y ) )
136, 12bitr4d 184 1  |-  ( ph  ->  ( Fun  F  <->  F :
( X /. R
) --> Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   {cab 2042   E.wrex 2324   _Vcvv 2574   <.cop 3406    |-> cmpt 3846   ran crn 4374   Fun wfun 4924   -->wf 4926    Er wer 6134   [cec 6135   /.cqs 6136
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-13 1420  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  ax-un 4198
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  df-er 6137  df-ec 6139  df-qs 6143
This theorem is referenced by: (None)
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