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Theorem fliftel 5808
Description: Elementhood in the relation  F. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
flift.1  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
flift.2  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
flift.3  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
Assertion
Ref Expression
fliftel  |-  ( ph  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B
) ) )
Distinct variable groups:    x, C    x, R    x, D    ph, x    x, X    x, S
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem fliftel
StepHypRef Expression
1 df-br 4024 . . 3  |-  ( C F D  <->  <. C ,  D >.  e.  F )
2 flift.1 . . . 4  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
32eleq2i 2347 . . 3  |-  ( <. C ,  D >.  e.  F  <->  <. C ,  D >.  e.  ran  ( x  e.  X  |->  <. A ,  B >. ) )
4 eqid 2283 . . . 4  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( x  e.  X  |->  <. A ,  B >. )
5 opex 4237 . . . 4  |-  <. A ,  B >.  e.  _V
64, 5elrnmpti 4930 . . 3  |-  ( <. C ,  D >.  e. 
ran  ( x  e.  X  |->  <. A ,  B >. )  <->  E. x  e.  X  <. C ,  D >.  = 
<. A ,  B >. )
71, 3, 63bitri 262 . 2  |-  ( C F D  <->  E. x  e.  X  <. C ,  D >.  =  <. A ,  B >. )
8 flift.2 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
9 flift.3 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
10 opthg2 4247 . . . 4  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( <. C ,  D >.  =  <. A ,  B >.  <-> 
( C  =  A  /\  D  =  B ) ) )
118, 9, 10syl2anc 642 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( <. C ,  D >.  = 
<. A ,  B >.  <->  ( C  =  A  /\  D  =  B )
) )
1211rexbidva 2560 . 2  |-  ( ph  ->  ( E. x  e.  X  <. C ,  D >.  =  <. A ,  B >.  <->  E. x  e.  X  ( C  =  A  /\  D  =  B
) ) )
137, 12syl5bb 248 1  |-  ( ph  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   <.cop 3643   class class class wbr 4023    e. cmpt 4077   ran crn 4690
This theorem is referenced by:  fliftcnv  5810  fliftfun  5811  fliftf  5814  fliftval  5815  qliftel  6741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-dm 4699  df-rn 4700
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