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Theorem fliftel 5554
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 3838 . . . 4  |-  ( C F D  <->  <. C ,  D >.  e.  F )
2 flift.1 . . . . 5  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
32eleq2i 2154 . . . 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 4046 . . . . . 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 2446 . . . 4  |-  ( ph  ->  A. x  e.  X  <. A ,  B >.  e. 
_V )
10 eqid 2088 . . . . 5  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( x  e.  X  |->  <. A ,  B >. )
1110elrnmptg 4675 . . . 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 4057 . . . 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 2377 . 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 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   _Vcvv 2619   <.cop 3444   class class class wbr 3837    |-> cmpt 3891   ran crn 4429
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-mpt 3893  df-cnv 4436  df-dm 4438  df-rn 4439
This theorem is referenced by:  fliftcnv  5556  fliftfun  5557  fliftf  5560  fliftval  5561  qliftel  6352
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