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Theorem dfon4 24504
Description: Another quantifier-free definition of  On. (Contributed by Scott Fenton, 4-May-2014.)
Assertion
Ref Expression
dfon4  |-  On  =  ( _V  \  (
( SSet  \  (  _I  u.  _E  ) )
" Trans ) )

Proof of Theorem dfon4
StepHypRef Expression
1 dfon3 24503 . 2  |-  On  =  ( _V  \  ran  (
( SSet  i^i  ( Trans  X.  _V ) ) 
\  (  _I  u.  _E  ) ) )
2 df-ima 4718 . . . 4  |-  ( (
SSet  \  (  _I  u.  _E  ) ) " Trans )  =  ran  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )
3 df-res 4717 . . . . . 6  |-  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ( ( SSet  \  (  _I  u.  _E  ) )  i^i  ( Trans  X.  _V ) )
4 indif1 3426 . . . . . 6  |-  ( (
SSet  \  (  _I  u.  _E  ) )  i^i  ( Trans  X.  _V ) )  =  ( ( SSet 
i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) )
53, 4eqtri 2316 . . . . 5  |-  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ( ( SSet  i^i  ( Trans  X.  _V )
)  \  (  _I  u.  _E  ) )
65rneqi 4921 . . . 4  |-  ran  (
( SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ran  ( ( SSet  i^i  ( Trans  X.  _V )
)  \  (  _I  u.  _E  ) )
72, 6eqtri 2316 . . 3  |-  ( (
SSet  \  (  _I  u.  _E  ) ) " Trans )  =  ran  ( (
SSet  i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) )
87difeq2i 3304 . 2  |-  ( _V 
\  ( ( SSet  \  (  _I  u.  _E  ) ) " Trans ) )  =  ( _V 
\  ran  ( ( SSet  i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) ) )
91, 8eqtr4i 2319 1  |-  On  =  ( _V  \  (
( SSet  \  (  _I  u.  _E  ) )
" Trans ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164    _E cep 4319    _I cid 4320   Oncon0 4408    X. cxp 4703   ran crn 4706    |` cres 4707   "cima 4708   SSetcsset 24446   Transctrans 24447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-txp 24466  df-sset 24468  df-trans 24469
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