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

Theorem fliftel1 5792
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 4227 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e. 
_V )
41, 2, 3syl2anc 411 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  _V )
5 eqid 2177 . . . . . 6  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( x  e.  X  |->  <. A ,  B >. )
65elrnmpt1 4877 . . . . 5  |-  ( ( x  e.  X  /\  <. A ,  B >.  e. 
_V )  ->  <. A ,  B >.  e.  ran  (
x  e.  X  |->  <. A ,  B >. ) )
76adantll 476 . . . 4  |-  ( ( ( ph  /\  x  e.  X )  /\  <. A ,  B >.  e.  _V )  ->  <. A ,  B >.  e.  ran  ( x  e.  X  |->  <. A ,  B >. ) )
84, 7mpdan 421 . . 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 2271 . 2  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  F )
11 df-br 4003 . 2  |-  ( A F B  <->  <. A ,  B >.  e.  F )
1210, 11sylibr 134 1  |-  ( (
ph  /\  x  e.  X )  ->  A F B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   _Vcvv 2737   <.cop 3595   class class class wbr 4002    |-> cmpt 4063   ran crn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-mpt 4065  df-cnv 4633  df-dm 4635  df-rn 4636
This theorem is referenced by:  fliftfun  5794  qliftel1  6613
  Copyright terms: Public domain W3C validator