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Theorem qliftlem 6626
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 6572 . . 3  |-  ( R  Er  X  ->  ( X  e.  _V  ->  R  e.  _V ) )
41, 2, 3sylc 62 . 2  |-  ( ph  ->  R  e.  _V )
5 ecelqsg 6601 . 2  |-  ( ( R  e.  _V  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R
) )
64, 5sylan 283 1  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1363    e. wcel 2158   _Vcvv 2749   <.cop 3607    |-> cmpt 4076   ran crn 4639    Er wer 6545   [cec 6546   /.cqs 6547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-er 6548  df-ec 6550  df-qs 6554
This theorem is referenced by:  qliftrel  6627  qliftel  6628  qliftel1  6629  qliftfun  6630  qliftf  6633  qliftval  6634
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