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Theorem qliftf 6746
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 6739 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
61, 5, 2fliftf 5814 . 2  |-  ( ph  ->  ( Fun  F  <->  F : ran  ( x  e.  X  |->  [ x ] R
) --> Y ) )
7 df-qs 6666 . . . . 5  |-  ( X /. R )  =  { y  |  E. x  e.  X  y  =  [ x ] R }
8 eqid 2283 . . . . . 6  |-  ( x  e.  X  |->  [ x ] R )  =  ( x  e.  X  |->  [ x ] R )
98rnmpt 4925 . . . . 5  |-  ran  (
x  e.  X  |->  [ x ] R )  =  { y  |  E. x  e.  X  y  =  [ x ] R }
107, 9eqtr4i 2306 . . . 4  |-  ( X /. R )  =  ran  ( x  e.  X  |->  [ x ] R )
1110a1i 10 . . 3  |-  ( ph  ->  ( X /. R
)  =  ran  (
x  e.  X  |->  [ x ] R ) )
1211feq2d 5380 . 2  |-  ( ph  ->  ( F : ( X /. R ) --> Y  <->  F : ran  (
x  e.  X  |->  [ x ] R ) --> Y ) )
136, 12bitr4d 247 1  |-  ( ph  ->  ( Fun  F  <->  F :
( X /. R
) --> Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   <.cop 3643    e. cmpt 4077   ran crn 4690   Fun wfun 5249   -->wf 5251    Er wer 6657   [cec 6658   /.cqs 6659
This theorem is referenced by:  orbsta  14767  frgpupf  15082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-er 6660  df-ec 6662  df-qs 6666
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