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Theorem fliftel 5660
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 3898 . . . 4  |-  ( C F D  <->  <. C ,  D >.  e.  F )
2 flift.1 . . . . 5  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
32eleq2i 2182 . . . 4  |-  ( <. C ,  D >.  e.  F  <->  <. C ,  D >.  e.  ran  ( x  e.  X  |->  <. A ,  B >. ) )
41, 3bitri 183 . . 3  |-  ( C F D  <->  <. C ,  D >.  e.  ran  (
x  e.  X  |->  <. A ,  B >. ) )
5 flift.2 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
6 flift.3 . . . . . 6  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
7 opexg 4118 . . . . . 6  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e. 
_V )
85, 6, 7syl2anc 406 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  _V )
98ralrimiva 2480 . . . 4  |-  ( ph  ->  A. x  e.  X  <. A ,  B >.  e. 
_V )
10 eqid 2115 . . . . 5  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( x  e.  X  |->  <. A ,  B >. )
1110elrnmptg 4759 . . . 4  |-  ( A. x  e.  X  <. A ,  B >.  e.  _V  ->  ( <. C ,  D >.  e.  ran  ( x  e.  X  |->  <. A ,  B >. )  <->  E. x  e.  X  <. C ,  D >.  =  <. A ,  B >. ) )
129, 11syl 14 . . 3  |-  ( ph  ->  ( <. C ,  D >.  e.  ran  ( x  e.  X  |->  <. A ,  B >. )  <->  E. x  e.  X  <. C ,  D >.  =  <. A ,  B >. ) )
134, 12syl5bb 191 . 2  |-  ( ph  ->  ( C F D  <->  E. x  e.  X  <. C ,  D >.  = 
<. A ,  B >. ) )
14 opthg2 4129 . . . 4  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( <. C ,  D >.  =  <. A ,  B >.  <-> 
( C  =  A  /\  D  =  B ) ) )
155, 6, 14syl2anc 406 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( <. C ,  D >.  = 
<. A ,  B >.  <->  ( C  =  A  /\  D  =  B )
) )
1615rexbidva 2409 . 2  |-  ( ph  ->  ( E. x  e.  X  <. C ,  D >.  =  <. A ,  B >.  <->  E. x  e.  X  ( C  =  A  /\  D  =  B
) ) )
1713, 16bitrd 187 1  |-  ( ph  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   A.wral 2391   E.wrex 2392   _Vcvv 2658   <.cop 3498   class class class wbr 3897    |-> cmpt 3957   ran crn 4508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-mpt 3959  df-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by:  fliftcnv  5662  fliftfun  5663  fliftf  5666  fliftval  5667  qliftel  6475
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