MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfsup3OLD Structured version   Unicode version

Theorem dfsup3OLD 7905
Description: Quantifier-free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dfsup3OLD  |-  sup ( B ,  A ,  R )  =  U. ( A  \  (
( `' R " B )  u.  ( R " ( A  \ 
( `' R " B ) ) ) ) )

Proof of Theorem dfsup3OLD
StepHypRef Expression
1 dfsup2OLD 7904 . 2  |-  sup ( B ,  A ,  R )  =  U. ( A  \  (
( `' R " B )  u.  (
( R  \  (
( `' R " B )  X.  _V ) ) " A
) ) )
2 indifcom 3743 . . . . . . . . 9  |-  ( ( A  X.  _V )  i^i  ( R  \  (
( `' R " B )  X.  _V ) ) )  =  ( R  i^i  (
( A  X.  _V )  \  ( ( `' R " B )  X.  _V ) ) )
3 incom 3691 . . . . . . . . 9  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )  =  ( ( A  X.  _V )  i^i  ( R  \ 
( ( `' R " B )  X.  _V ) ) )
4 difxp1 5432 . . . . . . . . . 10  |-  ( ( A  \  ( `' R " B ) )  X.  _V )  =  ( ( A  X.  _V )  \ 
( ( `' R " B )  X.  _V ) )
54ineq2i 3697 . . . . . . . . 9  |-  ( R  i^i  ( ( A 
\  ( `' R " B ) )  X. 
_V ) )  =  ( R  i^i  (
( A  X.  _V )  \  ( ( `' R " B )  X.  _V ) ) )
62, 3, 53eqtr4i 2506 . . . . . . . 8  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )  =  ( R  i^i  ( ( A  \  ( `' R " B ) )  X.  _V )
)
7 df-res 5011 . . . . . . . 8  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  |`  A )  =  ( ( R  \  (
( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )
8 df-res 5011 . . . . . . . 8  |-  ( R  |`  ( A  \  ( `' R " B ) ) )  =  ( R  i^i  ( ( A  \  ( `' R " B ) )  X.  _V )
)
96, 7, 83eqtr4i 2506 . . . . . . 7  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  |`  A )  =  ( R  |`  ( A  \  ( `' R " B ) ) )
109rneqi 5229 . . . . . 6  |-  ran  (
( R  \  (
( `' R " B )  X.  _V ) )  |`  A )  =  ran  ( R  |`  ( A  \  ( `' R " B ) ) )
11 df-ima 5012 . . . . . 6  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )
" A )  =  ran  ( ( R 
\  ( ( `' R " B )  X.  _V ) )  |`  A )
12 df-ima 5012 . . . . . 6  |-  ( R
" ( A  \ 
( `' R " B ) ) )  =  ran  ( R  |`  ( A  \  ( `' R " B ) ) )
1310, 11, 123eqtr4i 2506 . . . . 5  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )
" A )  =  ( R " ( A  \  ( `' R " B ) ) )
1413uneq2i 3655 . . . 4  |-  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) )  =  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) )
1514difeq2i 3619 . . 3  |-  ( A 
\  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) ) )  =  ( A 
\  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) ) )
1615unieqi 4254 . 2  |-  U. ( A  \  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) ) )  =  U. ( A  \  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) ) )
171, 16eqtri 2496 1  |-  sup ( B ,  A ,  R )  =  U. ( A  \  (
( `' R " B )  u.  ( R " ( A  \ 
( `' R " B ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475   U.cuni 4245    X. cxp 4997   `'ccnv 4998   ran crn 5000    |` cres 5001   "cima 5002   supcsup 7901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-sup 7902
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator