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Theorem relresfld 5524
Description: Restriction of a relation to its field. (Contributed by FL, 15-Apr-2012.)
Assertion
Ref Expression
relresfld  |-  ( Rel 
R  ->  ( R  |` 
U. U. R )  =  R )

Proof of Theorem relresfld
StepHypRef Expression
1 relfld 5523 . . . 4  |-  ( Rel 
R  ->  U. U. R  =  ( dom  R  u.  ran  R ) )
21reseq2d 5263 . . 3  |-  ( Rel 
R  ->  ( R  |` 
U. U. R )  =  ( R  |`  ( dom  R  u.  ran  R
) ) )
3 resundi 5277 . . 3  |-  ( R  |`  ( dom  R  u.  ran  R ) )  =  ( ( R  |`  dom  R )  u.  ( R  |`  ran  R ) )
4 eqtr 2469 . . . 4  |-  ( ( ( R  |`  U. U. R )  =  ( R  |`  ( dom  R  u.  ran  R ) )  /\  ( R  |`  ( dom  R  u.  ran  R ) )  =  ( ( R  |`  dom  R )  u.  ( R  |`  ran  R ) ) )  ->  ( R  |`  U. U. R
)  =  ( ( R  |`  dom  R )  u.  ( R  |`  ran  R ) ) )
5 resss 5287 . . . . 5  |-  ( R  |`  ran  R )  C_  R
6 resdm 5305 . . . . 5  |-  ( Rel 
R  ->  ( R  |` 
dom  R )  =  R )
7 ssequn2 3662 . . . . . 6  |-  ( ( R  |`  ran  R ) 
C_  R  <->  ( R  u.  ( R  |`  ran  R
) )  =  R )
8 uneq1 3636 . . . . . . . . 9  |-  ( ( R  |`  dom  R )  =  R  ->  (
( R  |`  dom  R
)  u.  ( R  |`  ran  R ) )  =  ( R  u.  ( R  |`  ran  R
) ) )
98eqeq2d 2457 . . . . . . . 8  |-  ( ( R  |`  dom  R )  =  R  ->  (
( R  |`  U. U. R )  =  ( ( R  |`  dom  R
)  u.  ( R  |`  ran  R ) )  <-> 
( R  |`  U. U. R )  =  ( R  u.  ( R  |`  ran  R ) ) ) )
10 eqtr 2469 . . . . . . . . 9  |-  ( ( ( R  |`  U. U. R )  =  ( R  u.  ( R  |`  ran  R ) )  /\  ( R  u.  ( R  |`  ran  R
) )  =  R )  ->  ( R  |` 
U. U. R )  =  R )
1110ex 434 . . . . . . . 8  |-  ( ( R  |`  U. U. R
)  =  ( R  u.  ( R  |`  ran  R ) )  -> 
( ( R  u.  ( R  |`  ran  R
) )  =  R  ->  ( R  |`  U.
U. R )  =  R ) )
129, 11syl6bi 228 . . . . . . 7  |-  ( ( R  |`  dom  R )  =  R  ->  (
( R  |`  U. U. R )  =  ( ( R  |`  dom  R
)  u.  ( R  |`  ran  R ) )  ->  ( ( R  u.  ( R  |`  ran  R ) )  =  R  ->  ( R  |` 
U. U. R )  =  R ) ) )
1312com3r 79 . . . . . 6  |-  ( ( R  u.  ( R  |`  ran  R ) )  =  R  ->  (
( R  |`  dom  R
)  =  R  -> 
( ( R  |`  U.
U. R )  =  ( ( R  |`  dom  R )  u.  ( R  |`  ran  R ) )  ->  ( R  |` 
U. U. R )  =  R ) ) )
147, 13sylbi 195 . . . . 5  |-  ( ( R  |`  ran  R ) 
C_  R  ->  (
( R  |`  dom  R
)  =  R  -> 
( ( R  |`  U.
U. R )  =  ( ( R  |`  dom  R )  u.  ( R  |`  ran  R ) )  ->  ( R  |` 
U. U. R )  =  R ) ) )
155, 6, 14mpsyl 63 . . . 4  |-  ( Rel 
R  ->  ( ( R  |`  U. U. R
)  =  ( ( R  |`  dom  R )  u.  ( R  |`  ran  R ) )  -> 
( R  |`  U. U. R )  =  R ) )
164, 15syl5com 30 . . 3  |-  ( ( ( R  |`  U. U. R )  =  ( R  |`  ( dom  R  u.  ran  R ) )  /\  ( R  |`  ( dom  R  u.  ran  R ) )  =  ( ( R  |`  dom  R )  u.  ( R  |`  ran  R ) ) )  ->  ( Rel  R  ->  ( R  |` 
U. U. R )  =  R ) )
172, 3, 16sylancl 662 . 2  |-  ( Rel 
R  ->  ( Rel  R  ->  ( R  |`  U.
U. R )  =  R ) )
1817pm2.43i 47 1  |-  ( Rel 
R  ->  ( R  |` 
U. U. R )  =  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    u. cun 3459    C_ wss 3461   U.cuni 4234   dom cdm 4989   ran crn 4990    |` cres 4991   Rel wrel 4994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001
This theorem is referenced by: (None)
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