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Theorem qliftel 6990
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 6988 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  [ x ] R  e.  ( X /. R ) )
61, 5, 2fliftel 6034 . 2  |-  ( ph  ->  ( [ C ] R F D  <->  E. x  e.  X  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
73adantr 453 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  R  Er  X )
8 simpr 449 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  x  e.  X )
97, 8erth2 6953 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  ( C R x  <->  [ C ] R  =  [
x ] R ) )
109anbi1d 687 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  (
( C R x  /\  D  =  A )  <->  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
1110rexbidva 2724 . 2  |-  ( ph  ->  ( E. x  e.  X  ( C R x  /\  D  =  A )  <->  E. x  e.  X  ( [ C ] R  =  [
x ] R  /\  D  =  A )
) )
126, 11bitr4d 249 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   _Vcvv 2958   <.cop 3819   class class class wbr 4215    e. cmpt 4269   ran crn 4882    Er wer 6905   [cec 6906   /.cqs 6907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-er 6908  df-ec 6910  df-qs 6914
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