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Theorem dfsup3OLD 7706
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 7705 . 2  |-  sup ( B ,  A ,  R )  =  U. ( A  \  (
( `' R " B )  u.  (
( R  \  (
( `' R " B )  X.  _V ) ) " A
) ) )
2 indifcom 3607 . . . . . . . . 9  |-  ( ( A  X.  _V )  i^i  ( R  \  (
( `' R " B )  X.  _V ) ) )  =  ( R  i^i  (
( A  X.  _V )  \  ( ( `' R " B )  X.  _V ) ) )
3 incom 3555 . . . . . . . . 9  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )  =  ( ( A  X.  _V )  i^i  ( R  \ 
( ( `' R " B )  X.  _V ) ) )
4 difxp1 5275 . . . . . . . . . 10  |-  ( ( A  \  ( `' R " B ) )  X.  _V )  =  ( ( A  X.  _V )  \ 
( ( `' R " B )  X.  _V ) )
54ineq2i 3561 . . . . . . . . 9  |-  ( R  i^i  ( ( A 
\  ( `' R " B ) )  X. 
_V ) )  =  ( R  i^i  (
( A  X.  _V )  \  ( ( `' R " B )  X.  _V ) ) )
62, 3, 53eqtr4i 2473 . . . . . . . 8  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )  =  ( R  i^i  ( ( A  \  ( `' R " B ) )  X.  _V )
)
7 df-res 4864 . . . . . . . 8  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  |`  A )  =  ( ( R  \  (
( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )
8 df-res 4864 . . . . . . . 8  |-  ( R  |`  ( A  \  ( `' R " B ) ) )  =  ( R  i^i  ( ( A  \  ( `' R " B ) )  X.  _V )
)
96, 7, 83eqtr4i 2473 . . . . . . 7  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  |`  A )  =  ( R  |`  ( A  \  ( `' R " B ) ) )
109rneqi 5078 . . . . . 6  |-  ran  (
( R  \  (
( `' R " B )  X.  _V ) )  |`  A )  =  ran  ( R  |`  ( A  \  ( `' R " B ) ) )
11 df-ima 4865 . . . . . 6  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )
" A )  =  ran  ( ( R 
\  ( ( `' R " B )  X.  _V ) )  |`  A )
12 df-ima 4865 . . . . . 6  |-  ( R
" ( A  \ 
( `' R " B ) ) )  =  ran  ( R  |`  ( A  \  ( `' R " B ) ) )
1310, 11, 123eqtr4i 2473 . . . . 5  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )
" A )  =  ( R " ( A  \  ( `' R " B ) ) )
1413uneq2i 3519 . . . 4  |-  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) )  =  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) )
1514difeq2i 3483 . . 3  |-  ( A 
\  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) ) )  =  ( A 
\  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) ) )
1615unieqi 4112 . 2  |-  U. ( A  \  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) ) )  =  U. ( A  \  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) ) )
171, 16eqtri 2463 1  |-  sup ( B ,  A ,  R )  =  U. ( A  \  (
( `' R " B )  u.  ( R " ( A  \ 
( `' R " B ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369   _Vcvv 2984    \ cdif 3337    u. cun 3338    i^i cin 3339   U.cuni 4103    X. cxp 4850   `'ccnv 4851   ran crn 4853    |` cres 4854   "cima 4855   supcsup 7702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-xp 4858  df-rel 4859  df-cnv 4860  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-sup 7703
This theorem is referenced by: (None)
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