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Theorem qliftel1 6561
Description: Elementhood in the relation  F. (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
qliftel1  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R F A )
Distinct variable groups:    ph, x    x, R    x, X    x, Y
Allowed substitution hints:    A( x)    F( x)

Proof of Theorem qliftel1
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 6558 . 2  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
61, 5, 2fliftel1 5744 1  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R F A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   _Vcvv 2712   <.cop 3563   class class class wbr 3965    |-> cmpt 4025   ran crn 4587    Er wer 6477   [cec 6478   /.cqs 6479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-xp 4592  df-rel 4593  df-cnv 4594  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-er 6480  df-ec 6482  df-qs 6486
This theorem is referenced by: (None)
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