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Theorem qliftlem 6250
Description:  F, a function lift, is a subset of  R  X.  S. (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
qliftlem  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
Distinct variable groups:    ph, x    x, R    x, X    x, Y
Allowed substitution hints:    A( x)    F( x)

Proof of Theorem qliftlem
StepHypRef Expression
1 qlift.3 . . 3  |-  ( ph  ->  R  Er  X )
2 qlift.4 . . 3  |-  ( ph  ->  X  e.  _V )
3 erex 6196 . . 3  |-  ( R  Er  X  ->  ( X  e.  _V  ->  R  e.  _V ) )
41, 2, 3sylc 61 . 2  |-  ( ph  ->  R  e.  _V )
5 ecelqsg 6225 . 2  |-  ( ( R  e.  _V  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R
) )
64, 5sylan 277 1  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   _Vcvv 2602   <.cop 3409    |-> cmpt 3847   ran crn 4372    Er wer 6169   [cec 6170   /.cqs 6171
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 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
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-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-er 6172  df-ec 6174  df-qs 6178
This theorem is referenced by:  qliftrel  6251  qliftel  6252  qliftel1  6253  qliftfun  6254  qliftf  6257  qliftval  6258
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