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Theorem dfon4 30660
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 30659 . 2  |-  On  =  ( _V  \  ran  (
( SSet  i^i  ( Trans  X.  _V ) ) 
\  (  _I  u.  _E  ) ) )
2 df-ima 4847 . . . 4  |-  ( (
SSet  \  (  _I  u.  _E  ) ) " Trans )  =  ran  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )
3 df-res 4846 . . . . . 6  |-  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ( ( SSet  \  (  _I  u.  _E  ) )  i^i  ( Trans  X.  _V ) )
4 indif1 3687 . . . . . 6  |-  ( (
SSet  \  (  _I  u.  _E  ) )  i^i  ( Trans  X.  _V ) )  =  ( ( SSet 
i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) )
53, 4eqtri 2473 . . . . 5  |-  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ( ( SSet  i^i  ( Trans  X.  _V )
)  \  (  _I  u.  _E  ) )
65rneqi 5061 . . . 4  |-  ran  (
( SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ran  ( ( SSet  i^i  ( Trans  X.  _V )
)  \  (  _I  u.  _E  ) )
72, 6eqtri 2473 . . 3  |-  ( (
SSet  \  (  _I  u.  _E  ) ) " Trans )  =  ran  ( (
SSet  i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) )
87difeq2i 3548 . 2  |-  ( _V 
\  ( ( SSet  \  (  _I  u.  _E  ) ) " Trans ) )  =  ( _V 
\  ran  ( ( SSet  i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) ) )
91, 8eqtr4i 2476 1  |-  On  =  ( _V  \  (
( SSet  \  (  _I  u.  _E  ) )
" Trans ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1444   _Vcvv 3045    \ cdif 3401    u. cun 3402    i^i cin 3403    _E cep 4743    _I cid 4744    X. cxp 4832   ran crn 4835    |` cres 4836   "cima 4837   Oncon0 5423   SSetcsset 30598   Transctrans 30599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fo 5588  df-fv 5590  df-1st 6793  df-2nd 6794  df-txp 30620  df-sset 30622  df-trans 30623
This theorem is referenced by: (None)
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