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Theorem dfsup3OLD 7197
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 7196 . 2  |-  sup ( B ,  A ,  R )  =  U. ( A  \  (
( `' R " B )  u.  (
( R  \  (
( `' R " B )  X.  _V ) ) " A
) ) )
2 indifcom 3414 . . . . . . . . 9  |-  ( ( A  X.  _V )  i^i  ( R  \  (
( `' R " B )  X.  _V ) ) )  =  ( R  i^i  (
( A  X.  _V )  \  ( ( `' R " B )  X.  _V ) ) )
3 incom 3361 . . . . . . . . 9  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )  =  ( ( A  X.  _V )  i^i  ( R  \ 
( ( `' R " B )  X.  _V ) ) )
4 difxp1 6154 . . . . . . . . . 10  |-  ( ( A  \  ( `' R " B ) )  X.  _V )  =  ( ( A  X.  _V )  \ 
( ( `' R " B )  X.  _V ) )
54ineq2i 3367 . . . . . . . . 9  |-  ( R  i^i  ( ( A 
\  ( `' R " B ) )  X. 
_V ) )  =  ( R  i^i  (
( A  X.  _V )  \  ( ( `' R " B )  X.  _V ) ) )
62, 3, 53eqtr4i 2313 . . . . . . . 8  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )  =  ( R  i^i  ( ( A  \  ( `' R " B ) )  X.  _V )
)
7 df-res 4701 . . . . . . . 8  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  |`  A )  =  ( ( R  \  (
( `' R " B )  X.  _V ) )  i^i  ( A  X.  _V ) )
8 df-res 4701 . . . . . . . 8  |-  ( R  |`  ( A  \  ( `' R " B ) ) )  =  ( R  i^i  ( ( A  \  ( `' R " B ) )  X.  _V )
)
96, 7, 83eqtr4i 2313 . . . . . . 7  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )  |`  A )  =  ( R  |`  ( A  \  ( `' R " B ) ) )
109rneqi 4905 . . . . . 6  |-  ran  (
( R  \  (
( `' R " B )  X.  _V ) )  |`  A )  =  ran  ( R  |`  ( A  \  ( `' R " B ) ) )
11 df-ima 4702 . . . . . 6  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )
" A )  =  ran  ( ( R 
\  ( ( `' R " B )  X.  _V ) )  |`  A )
12 df-ima 4702 . . . . . 6  |-  ( R
" ( A  \ 
( `' R " B ) ) )  =  ran  ( R  |`  ( A  \  ( `' R " B ) ) )
1310, 11, 123eqtr4i 2313 . . . . 5  |-  ( ( R  \  ( ( `' R " B )  X.  _V ) )
" A )  =  ( R " ( A  \  ( `' R " B ) ) )
1413uneq2i 3326 . . . 4  |-  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) )  =  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) )
1514difeq2i 3291 . . 3  |-  ( A 
\  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) ) )  =  ( A 
\  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) ) )
1615unieqi 3837 . 2  |-  U. ( A  \  ( ( `' R " B )  u.  ( ( R 
\  ( ( `' R " B )  X.  _V ) )
" A ) ) )  =  U. ( A  \  ( ( `' R " B )  u.  ( R "
( A  \  ( `' R " B ) ) ) ) )
171, 16eqtri 2303 1  |-  sup ( B ,  A ,  R )  =  U. ( A  \  (
( `' R " B )  u.  ( R " ( A  \ 
( `' R " B ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151   U.cuni 3827    X. cxp 4687   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692   supcsup 7193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-sup 7194
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