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Theorem dfsup3OLD 7451
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 7450 . 2  |-  sup ( B ,  A ,  R )  =  U. ( A  \  (
( `' R " B )  u.  (
( R  \  (
( `' R " B )  X.  _V ) ) " A
) ) )
2 indifcom 3588 . . . . . . . . 9  |-  ( ( A  X.  _V )  i^i  ( R  \  (
( `' R " B )  X.  _V ) ) )  =  ( R  i^i  (
( A  X.  _V )  \  ( ( `' R " B )  X.  _V ) ) )
3 incom 3535 . . . . . . . . 9  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )  =  ( ( A  X.  _V )  i^i  ( R  \ 
( ( `' R " B )  X.  _V ) ) )
4 difxp1 6383 . . . . . . . . . 10  |-  ( ( A  \  ( `' R " B ) )  X.  _V )  =  ( ( A  X.  _V )  \ 
( ( `' R " B )  X.  _V ) )
54ineq2i 3541 . . . . . . . . 9  |-  ( R  i^i  ( ( A 
\  ( `' R " B ) )  X. 
_V ) )  =  ( R  i^i  (
( A  X.  _V )  \  ( ( `' R " B )  X.  _V ) ) )
62, 3, 53eqtr4i 2468 . . . . . . . 8  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )  =  ( R  i^i  ( ( A  \  ( `' R " B ) )  X.  _V )
)
7 df-res 4892 . . . . . . . 8  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  |`  A )  =  ( ( R  \  (
( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )
8 df-res 4892 . . . . . . . 8  |-  ( R  |`  ( A  \  ( `' R " B ) ) )  =  ( R  i^i  ( ( A  \  ( `' R " B ) )  X.  _V )
)
96, 7, 83eqtr4i 2468 . . . . . . 7  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  |`  A )  =  ( R  |`  ( A  \  ( `' R " B ) ) )
109rneqi 5098 . . . . . 6  |-  ran  (
( R  \  (
( `' R " B )  X.  _V ) )  |`  A )  =  ran  ( R  |`  ( A  \  ( `' R " B ) ) )
11 df-ima 4893 . . . . . 6  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )
" A )  =  ran  ( ( R 
\  ( ( `' R " B )  X.  _V ) )  |`  A )
12 df-ima 4893 . . . . . 6  |-  ( R
" ( A  \ 
( `' R " B ) ) )  =  ran  ( R  |`  ( A  \  ( `' R " B ) ) )
1310, 11, 123eqtr4i 2468 . . . . 5  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )
" A )  =  ( R " ( A  \  ( `' R " B ) ) )
1413uneq2i 3500 . . . 4  |-  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) )  =  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) )
1514difeq2i 3464 . . 3  |-  ( A 
\  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) ) )  =  ( A 
\  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) ) )
1615unieqi 4027 . 2  |-  U. ( A  \  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) ) )  =  U. ( A  \  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) ) )
171, 16eqtri 2458 1  |-  sup ( B ,  A ,  R )  =  U. ( A  \  (
( `' R " B )  u.  ( R " ( A  \ 
( `' R " B ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653   _Vcvv 2958    \ cdif 3319    u. cun 3320    i^i cin 3321   U.cuni 4017    X. cxp 4878   `'ccnv 4879   ran crn 4881    |` cres 4882   "cima 4883   supcsup 7447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-sup 7448
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