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Theorem qliftel 6274
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 6272 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
61, 5, 2fliftel 5485 . 2  |-  ( ph  ->  ( [ C ] R F D  <->  E. x  e.  X  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
73adantr 270 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  R  Er  X )
8 simpr 108 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
97, 8erth2 6239 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( C R x  <->  [ C ] R  =  [
x ] R ) )
109anbi1d 453 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( C R x  /\  D  =  A )  <->  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
1110rexbidva 2370 . 2  |-  ( ph  ->  ( E. x  e.  X  ( C R x  /\  D  =  A )  <->  E. x  e.  X  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
126, 11bitr4d 189 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 102    <-> wb 103    = wceq 1285    e. wcel 1434   E.wrex 2354   _Vcvv 2610   <.cop 3420   class class class wbr 3806    |-> cmpt 3860   ran crn 4393    Er wer 6191   [cec 6192   /.cqs 6193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2826  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-mpt 3862  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-er 6194  df-ec 6196  df-qs 6200
This theorem is referenced by: (None)
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