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Theorem fliftel1 5747
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
fliftel1  |-  ( (
ph  /\  x  e.  X )  ->  A F B )
Distinct variable groups:    x, R    ph, x    x, X    x, S
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem fliftel1
StepHypRef Expression
1 flift.2 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
2 flift.3 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
3 opexg 4191 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e. 
_V )
41, 2, 3syl2anc 409 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  _V )
5 eqid 2157 . . . . . 6  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( x  e.  X  |->  <. A ,  B >. )
65elrnmpt1 4840 . . . . 5  |-  ( ( x  e.  X  /\  <. A ,  B >.  e. 
_V )  ->  <. A ,  B >.  e.  ran  (
x  e.  X  |->  <. A ,  B >. ) )
76adantll 468 . . . 4  |-  ( ( ( ph  /\  x  e.  X )  /\  <. A ,  B >.  e.  _V )  ->  <. A ,  B >.  e.  ran  ( x  e.  X  |->  <. A ,  B >. ) )
84, 7mpdan 418 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  ran  (
x  e.  X  |->  <. A ,  B >. ) )
9 flift.1 . . 3  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
108, 9eleqtrrdi 2251 . 2  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  F )
11 df-br 3968 . 2  |-  ( A F B  <->  <. A ,  B >.  e.  F )
1210, 11sylibr 133 1  |-  ( (
ph  /\  x  e.  X )  ->  A F B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   _Vcvv 2712   <.cop 3564   class class class wbr 3967    |-> cmpt 4028   ran crn 4590
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4029  df-mpt 4030  df-cnv 4597  df-dm 4599  df-rn 4600
This theorem is referenced by:  fliftfun  5749  qliftel1  6564
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