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Theorem fliftel 5464
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 3794 . . . 4  |-  ( C F D  <->  <. C ,  D >.  e.  F )
2 flift.1 . . . . 5  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
32eleq2i 2146 . . . 4  |-  ( <. C ,  D >.  e.  F  <->  <. C ,  D >.  e.  ran  ( x  e.  X  |->  <. A ,  B >. ) )
41, 3bitri 182 . . 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 3991 . . . . . 6  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e. 
_V )
85, 6, 7syl2anc 403 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  _V )
98ralrimiva 2435 . . . 4  |-  ( ph  ->  A. x  e.  X  <. A ,  B >.  e. 
_V )
10 eqid 2082 . . . . 5  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( x  e.  X  |->  <. A ,  B >. )
1110elrnmptg 4614 . . . 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 190 . 2  |-  ( ph  ->  ( C F D  <->  E. x  e.  X  <. C ,  D >.  = 
<. A ,  B >. ) )
14 opthg2 4002 . . . 4  |-  ( ( A  e.  R  /\  B  e.  S )  ->  ( <. C ,  D >.  =  <. A ,  B >.  <-> 
( C  =  A  /\  D  =  B ) ) )
155, 6, 14syl2anc 403 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  ( <. C ,  D >.  = 
<. A ,  B >.  <->  ( C  =  A  /\  D  =  B )
) )
1615rexbidva 2366 . 2  |-  ( ph  ->  ( E. x  e.  X  <. C ,  D >.  =  <. A ,  B >.  <->  E. x  e.  X  ( C  =  A  /\  D  =  B
) ) )
1713, 16bitrd 186 1  |-  ( ph  ->  ( C F D  <->  E. x  e.  X  ( C  =  A  /\  D  =  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   A.wral 2349   E.wrex 2350   _Vcvv 2602   <.cop 3409   class class class wbr 3793    |-> cmpt 3847   ran crn 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-mpt 3849  df-cnv 4379  df-dm 4381  df-rn 4382
This theorem is referenced by:  fliftcnv  5466  fliftfun  5467  fliftf  5470  fliftval  5471  qliftel  6252
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