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

Theorem fliftrel 5571
Description:  F, a function lift, is a subset of  R  X.  S. (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
fliftrel  |-  ( ph  ->  F  C_  ( R  X.  S ) )
Distinct variable groups:    x, R    ph, x    x, X    x, S
Allowed substitution hints:    A( x)    B( x)    F( x)

Proof of Theorem fliftrel
StepHypRef Expression
1 flift.1 . 2  |-  F  =  ran  ( x  e.  X  |->  <. A ,  B >. )
2 flift.2 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  A  e.  R )
3 flift.3 . . . . 5  |-  ( (
ph  /\  x  e.  X )  ->  B  e.  S )
4 opelxpi 4469 . . . . 5  |-  ( ( A  e.  R  /\  B  e.  S )  -> 
<. A ,  B >.  e.  ( R  X.  S
) )
52, 3, 4syl2anc 403 . . . 4  |-  ( (
ph  /\  x  e.  X )  ->  <. A ,  B >.  e.  ( R  X.  S ) )
6 eqid 2088 . . . 4  |-  ( x  e.  X  |->  <. A ,  B >. )  =  ( x  e.  X  |->  <. A ,  B >. )
75, 6fmptd 5452 . . 3  |-  ( ph  ->  ( x  e.  X  |-> 
<. A ,  B >. ) : X --> ( R  X.  S ) )
8 frn 5169 . . 3  |-  ( ( x  e.  X  |->  <. A ,  B >. ) : X --> ( R  X.  S )  ->  ran  ( x  e.  X  |-> 
<. A ,  B >. ) 
C_  ( R  X.  S ) )
97, 8syl 14 . 2  |-  ( ph  ->  ran  ( x  e.  X  |->  <. A ,  B >. )  C_  ( R  X.  S ) )
101, 9syl5eqss 3070 1  |-  ( ph  ->  F  C_  ( R  X.  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438    C_ wss 2999   <.cop 3449    |-> cmpt 3899    X. cxp 4436   ran crn 4439   -->wf 5011
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 3957  ax-pow 4009  ax-pr 4036
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-rab 2368  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023
This theorem is referenced by:  fliftcnv  5574  fliftfun  5575  fliftf  5578  qliftrel  6369
  Copyright terms: Public domain W3C validator