ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fliftel1 Unicode version

Theorem fliftel1 5421
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 3961 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e. 
_V )
41, 2, 3syl2anc 391 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  _V )
5 eqid 2040 . . . . . 6  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( x  e.  X  |->  <. A ,  B >. )
65elrnmpt1 4572 . . . . 5  |-  ( ( x  e.  X  /\  <. A ,  B >.  e. 
_V )  ->  <. A ,  B >.  e.  ran  (
x  e.  X  |->  <. A ,  B >. ) )
76adantll 445 . . . 4  |-  ( ( ( ph  /\  x  e.  X )  /\  <. A ,  B >.  e.  _V )  ->  <. A ,  B >.  e.  ran  ( x  e.  X  |->  <. A ,  B >. ) )
84, 7mpdan 398 . . 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, 9syl6eleqr 2131 . 2  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  F )
11 df-br 3762 . 2  |-  ( A F B  <->  <. A ,  B >.  e.  F )
1210, 11sylibr 137 1  |-  ( (
ph  /\  x  e.  X )  ->  A F B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   _Vcvv 2554   <.cop 3375   class class class wbr 3761    |-> cmpt 3815   ran crn 4333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2309  df-v 2556  df-sbc 2762  df-csb 2850  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-br 3762  df-opab 3816  df-mpt 3817  df-cnv 4340  df-dm 4342  df-rn 4343
This theorem is referenced by:  fliftfun  5423  qliftel1  6174
  Copyright terms: Public domain W3C validator