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Theorem qliftel 6829
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 6827 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
61, 5, 2fliftel 5895 . 2  |-  ( ph  ->  ( [ C ] R F D  <->  E. x  e.  X  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
73adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  R  Er  X )
8 simpr 447 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
97, 8erth2 6792 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( C R x  <->  [ C ] R  =  [
x ] R ) )
109anbi1d 685 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( C R x  /\  D  =  A )  <->  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
1110rexbidva 2636 . 2  |-  ( ph  ->  ( E. x  e.  X  ( C R x  /\  D  =  A )  <->  E. x  e.  X  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
126, 11bitr4d 247 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    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   E.wrex 2620   _Vcvv 2864   <.cop 3719   class class class wbr 4104    e. cmpt 4158   ran crn 4772    Er wer 6744   [cec 6745   /.cqs 6746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-er 6747  df-ec 6749  df-qs 6753
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