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Theorem qliftlem 6437
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 6383 . . 3  |-  ( R  Er  X  ->  ( X  e.  _V  ->  R  e.  _V ) )
41, 2, 3sylc 62 . 2  |-  ( ph  ->  R  e.  _V )
5 ecelqsg 6412 . 2  |-  ( ( R  e.  _V  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R
) )
64, 5sylan 279 1  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1299    e. wcel 1448   _Vcvv 2641   <.cop 3477    |-> cmpt 3929   ran crn 4478    Er wer 6356   [cec 6357   /.cqs 6358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-xp 4483  df-rel 4484  df-cnv 4485  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-er 6359  df-ec 6361  df-qs 6365
This theorem is referenced by:  qliftrel  6438  qliftel  6439  qliftel1  6440  qliftfun  6441  qliftf  6444  qliftval  6445
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