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Theorem qliftlem 6977
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 6921 . . 3  |-  ( R  Er  X  ->  ( X  e.  _V  ->  R  e.  _V ) )
41, 2, 3sylc 58 . 2  |-  ( ph  ->  R  e.  _V )
5 ecelqsg 6951 . 2  |-  ( ( R  e.  _V  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R
) )
64, 5sylan 458 1  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809    e. cmpt 4258   ran crn 4871    Er wer 6894   [cec 6895   /.cqs 6896
This theorem is referenced by:  qliftrel  6978  qliftel  6979  qliftel1  6980  qliftfun  6981  qliftf  6984  qliftval  6985
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-er 6897  df-ec 6899  df-qs 6903
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