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Theorem qliftrel 6783
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
qliftrel  |-  ( ph  ->  F  C_  ( ( X /. R )  X.  Y ) )
Distinct variable groups:    ph, x    x, R    x, X    x, Y
Allowed substitution hints:    A( x)    F( x)

Proof of Theorem qliftrel
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 6782 . 2  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
61, 5, 2fliftrel 5849 1  |-  ( ph  ->  F  C_  ( ( X /. R )  X.  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   _Vcvv 2822    C_ wss 3186   <.cop 3677    e. cmpt 4114    X. cxp 4724   ran crn 4727    Er wer 6699   [cec 6700   /.cqs 6701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-er 6702  df-ec 6704  df-qs 6708
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