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Theorem fliftel1 6032
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 opex 4427 . . . . 5  |-  <. A ,  B >.  e.  _V
2 eqid 2436 . . . . . 6  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( x  e.  X  |->  <. A ,  B >. )
32elrnmpt1 5119 . . . . 5  |-  ( ( x  e.  X  /\  <. A ,  B >.  e. 
_V )  ->  <. A ,  B >.  e.  ran  (
x  e.  X  |->  <. A ,  B >. ) )
41, 3mpan2 653 . . . 4  |-  ( x  e.  X  ->  <. A ,  B >.  e.  ran  (
x  e.  X  |->  <. A ,  B >. ) )
54adantl 453 . . 3  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  ran  (
x  e.  X  |->  <. A ,  B >. ) )
6 flift.1 . . 3  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
75, 6syl6eleqr 2527 . 2  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  F )
8 df-br 4213 . 2  |-  ( A F B  <->  <. A ,  B >.  e.  F )
97, 8sylibr 204 1  |-  ( (
ph  /\  x  e.  X )  ->  A F B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817   class class class wbr 4212    e. cmpt 4266   ran crn 4879
This theorem is referenced by:  fliftfun  6034  qliftel1  6988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-mpt 4268  df-cnv 4886  df-dm 4888  df-rn 4889
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