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Theorem qliftel 6641
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
qliftel  |-  ( ph  ->  ( [ C ] R F D  <->  E. x  e.  X  ( C R x  /\  D  =  A ) ) )
Distinct variable groups:    x, C    x, D    ph, x    x, R    x, X    x, Y
Allowed substitution hints:    A( x)    F( x)

Proof of Theorem qliftel
StepHypRef Expression
1 qlift.1 . . 3  |-  F  =  ran  ( x  e.  X  |->  <. [ x ] R ,  A >. )
2 qlift.2 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  Y )
3 qlift.3 . . . 4  |-  ( ph  ->  R  Er  X )
4 qlift.4 . . . 4  |-  ( ph  ->  X  e.  _V )
51, 2, 3, 4qliftlem 6639 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
61, 5, 2fliftel 5815 . 2  |-  ( ph  ->  ( [ C ] R F D  <->  E. x  e.  X  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
73adantr 276 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  R  Er  X )
8 simpr 110 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
97, 8erth2 6606 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( C R x  <->  [ C ] R  =  [
x ] R ) )
109anbi1d 465 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( C R x  /\  D  =  A )  <->  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
1110rexbidva 2487 . 2  |-  ( ph  ->  ( E. x  e.  X  ( C R x  /\  D  =  A )  <->  E. x  e.  X  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
126, 11bitr4d 191 1  |-  ( ph  ->  ( [ C ] R F D  <->  E. x  e.  X  ( C R x  /\  D  =  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   E.wrex 2469   _Vcvv 2752   <.cop 3610   class class class wbr 4018    |-> cmpt 4079   ran crn 4645    Er wer 6556   [cec 6557   /.cqs 6558
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-er 6559  df-ec 6561  df-qs 6565
This theorem is referenced by: (None)
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