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Theorem qliftval 6522
Description: The value 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 )
qliftval.4  |-  ( x  =  C  ->  A  =  B )
qliftval.6  |-  ( ph  ->  Fun  F )
Assertion
Ref Expression
qliftval  |-  ( (
ph  /\  C  e.  X )  ->  ( F `  [ C ] R )  =  B )
Distinct variable groups:    x, B    x, C    ph, x    x, R    x, X    x, Y
Allowed substitution hints:    A( x)    F( x)

Proof of Theorem qliftval
StepHypRef Expression
1 qlift.1 . 2  |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )
2 qlift.2 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Y )
3 qlift.3 . . 3  |-  ( ph  ->  R  Er  X )
4 qlift.4 . . 3  |-  ( ph  ->  X  e.  _V )
51, 2, 3, 4qliftlem 6514 . 2  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
6 eceq1 6471 . 2  |-  ( x  =  C  ->  [ x ] R  =  [ C ] R )
7 qliftval.4 . 2  |-  ( x  =  C  ->  A  =  B )
8 qliftval.6 . 2  |-  ( ph  ->  Fun  F )
91, 5, 2, 6, 7, 8fliftval 5708 1  |-  ( (
ph  /\  C  e.  X )  ->  ( F `  [ C ] R )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332    e. wcel 1481   _Vcvv 2689   <.cop 3534    |-> cmpt 3996   ran crn 4547   Fun wfun 5124   ` cfv 5130    Er wer 6433   [cec 6434   /.cqs 6435
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fv 5138  df-er 6436  df-ec 6438  df-qs 6442
This theorem is referenced by: (None)
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