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

Theorem dfsup3OLD 7242
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 7241 . 2  |-  sup ( B ,  A ,  R )  =  U. ( A  \  (
( `' R " B )  u.  (
( R  \  (
( `' R " B )  X.  _V ) ) " A
) ) )
2 indifcom 3448 . . . . . . . . 9  |-  ( ( A  X.  _V )  i^i  ( R  \  (
( `' R " B )  X.  _V ) ) )  =  ( R  i^i  (
( A  X.  _V )  \  ( ( `' R " B )  X.  _V ) ) )
3 incom 3395 . . . . . . . . 9  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )  =  ( ( A  X.  _V )  i^i  ( R  \ 
( ( `' R " B )  X.  _V ) ) )
4 difxp1 6196 . . . . . . . . . 10  |-  ( ( A  \  ( `' R " B ) )  X.  _V )  =  ( ( A  X.  _V )  \ 
( ( `' R " B )  X.  _V ) )
54ineq2i 3401 . . . . . . . . 9  |-  ( R  i^i  ( ( A 
\  ( `' R " B ) )  X. 
_V ) )  =  ( R  i^i  (
( A  X.  _V )  \  ( ( `' R " B )  X.  _V ) ) )
62, 3, 53eqtr4i 2346 . . . . . . . 8  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )  =  ( R  i^i  ( ( A  \  ( `' R " B ) )  X.  _V )
)
7 df-res 4738 . . . . . . . 8  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  |`  A )  =  ( ( R  \  (
( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )
8 df-res 4738 . . . . . . . 8  |-  ( R  |`  ( A  \  ( `' R " B ) ) )  =  ( R  i^i  ( ( A  \  ( `' R " B ) )  X.  _V )
)
96, 7, 83eqtr4i 2346 . . . . . . 7  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  |`  A )  =  ( R  |`  ( A  \  ( `' R " B ) ) )
109rneqi 4942 . . . . . 6  |-  ran  (
( R  \  (
( `' R " B )  X.  _V ) )  |`  A )  =  ran  ( R  |`  ( A  \  ( `' R " B ) ) )
11 df-ima 4739 . . . . . 6  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )
" A )  =  ran  ( ( R 
\  ( ( `' R " B )  X.  _V ) )  |`  A )
12 df-ima 4739 . . . . . 6  |-  ( R
" ( A  \ 
( `' R " B ) ) )  =  ran  ( R  |`  ( A  \  ( `' R " B ) ) )
1310, 11, 123eqtr4i 2346 . . . . 5  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )
" A )  =  ( R " ( A  \  ( `' R " B ) ) )
1413uneq2i 3360 . . . 4  |-  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) )  =  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) )
1514difeq2i 3325 . . 3  |-  ( A 
\  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) ) )  =  ( A 
\  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) ) )
1615unieqi 3874 . 2  |-  U. ( A  \  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) ) )  =  U. ( A  \  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) ) )
171, 16eqtri 2336 1  |-  sup ( B ,  A ,  R )  =  U. ( A  \  (
( `' R " B )  u.  ( R " ( A  \ 
( `' R " B ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1633   _Vcvv 2822    \ cdif 3183    u. cun 3184    i^i cin 3185   U.cuni 3864    X. cxp 4724   `'ccnv 4725   ran crn 4727    |` cres 4728   "cima 4729   supcsup 7238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-xp 4732  df-rel 4733  df-cnv 4734  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-sup 7239
  Copyright terms: Public domain W3C validator