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Theorem fliftel 6033
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 4215 . . 3  |-  ( C F D  <->  <. C ,  D >.  e.  F )
2 flift.1 . . . 4  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
32eleq2i 2502 . . 3  |-  ( <. C ,  D >.  e.  F  <->  <. C ,  D >.  e.  ran  ( x  e.  X  |->  <. A ,  B >. ) )
4 eqid 2438 . . . 4  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( x  e.  X  |->  <. A ,  B >. )
5 opex 4429 . . . 4  |-  <. A ,  B >.  e.  _V
64, 5elrnmpti 5123 . . 3  |-  ( <. C ,  D >.  e. 
ran  ( x  e.  X  |->  <. A ,  B >. )  <->  E. x  e.  X  <. C ,  D >.  = 
<. A ,  B >. )
71, 3, 63bitri 264 . 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 4439 . . . 4  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( <. C ,  D >.  =  <. A ,  B >.  <-> 
( C  =  A  /\  D  =  B ) ) )
118, 9, 10syl2anc 644 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( <. C ,  D >.  = 
<. A ,  B >.  <->  ( C  =  A  /\  D  =  B )
) )
1211rexbidva 2724 . 2  |-  ( ph  ->  ( E. x  e.  X  <. C ,  D >.  =  <. A ,  B >.  <->  E. x  e.  X  ( C  =  A  /\  D  =  B
) ) )
137, 12syl5bb 250 1  |-  ( ph  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2708   <.cop 3819   class class class wbr 4214    e. cmpt 4268   ran crn 4881
This theorem is referenced by:  fliftcnv  6035  fliftfun  6036  fliftf  6039  fliftval  6040  qliftel  6989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
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-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-mpt 4270  df-cnv 4888  df-dm 4890  df-rn 4891
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