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Theorem qliftval 6251
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 6243 . 2  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
6 eceq1 6200 . 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 5465 1  |-  ( (
ph  /\  C  e.  X )  ->  ( F `  [ C ] R )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2602   <.cop 3403    |-> cmpt 3841   ran crn 4366   Fun wfun 4920   ` cfv 4926    Er wer 6162   [cec 6163   /.cqs 6164
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fv 4934  df-er 6165  df-ec 6167  df-qs 6171
This theorem is referenced by: (None)
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