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Theorem qliftel 7412
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 7410 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
61, 5, 2fliftel 6208 . 2  |-  ( ph  ->  ( [ C ] R F D  <->  E. x  e.  X  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
73adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  R  Er  X )
8 simpr 461 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
97, 8erth2 7375 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( C R x  <->  [ C ] R  =  [
x ] R ) )
109anbi1d 704 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( C R x  /\  D  =  A )  <->  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
1110rexbidva 2965 . 2  |-  ( ph  ->  ( E. x  e.  X  ( C R x  /\  D  =  A )  <->  E. x  e.  X  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
126, 11bitr4d 256 1  |-  ( ph  ->  ( [ C ] R F D  <->  E. x  e.  X  ( C R x  /\  D  =  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109   <.cop 4038   class class class wbr 4456    |-> cmpt 4515   ran crn 5009    Er wer 7326   [cec 7327   /.cqs 7328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-er 7329  df-ec 7331  df-qs 7335
This theorem is referenced by: (None)
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