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Theorem qliftf 6562
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 6555 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
61, 5, 2fliftf 5746 . 2  |-  ( ph  ->  ( Fun  F  <->  F : ran  ( x  e.  X  |->  [ x ] R
) --> Y ) )
7 df-qs 6483 . . . . 5  |-  ( X /. R )  =  { y  |  E. x  e.  X  y  =  [ x ] R }
8 eqid 2157 . . . . . 6  |-  ( x  e.  X  |->  [ x ] R )  =  ( x  e.  X  |->  [ x ] R )
98rnmpt 4833 . . . . 5  |-  ran  (
x  e.  X  |->  [ x ] R )  =  { y  |  E. x  e.  X  y  =  [ x ] R }
107, 9eqtr4i 2181 . . . 4  |-  ( X /. R )  =  ran  ( x  e.  X  |->  [ x ] R )
1110a1i 9 . . 3  |-  ( ph  ->  ( X /. R
)  =  ran  (
x  e.  X  |->  [ x ] R ) )
1211feq2d 5306 . 2  |-  ( ph  ->  ( F : ( X /. R ) --> Y  <->  F : ran  (
x  e.  X  |->  [ x ] R ) --> Y ) )
136, 12bitr4d 190 1  |-  ( ph  ->  ( Fun  F  <->  F :
( X /. R
) --> Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   {cab 2143   E.wrex 2436   _Vcvv 2712   <.cop 3563    |-> cmpt 4025   ran crn 4586   Fun wfun 5163   -->wf 5165    Er wer 6474   [cec 6475   /.cqs 6476
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-fv 5177  df-er 6477  df-ec 6479  df-qs 6483
This theorem is referenced by: (None)
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