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Theorem dfon4 28091
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 28090 . 2  |-  On  =  ( _V  \  ran  (
( SSet  i^i  ( Trans  X.  _V ) ) 
\  (  _I  u.  _E  ) ) )
2 df-ima 4964 . . . 4  |-  ( (
SSet  \  (  _I  u.  _E  ) ) " Trans )  =  ran  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )
3 df-res 4963 . . . . . 6  |-  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ( ( SSet  \  (  _I  u.  _E  ) )  i^i  ( Trans  X.  _V ) )
4 indif1 3705 . . . . . 6  |-  ( (
SSet  \  (  _I  u.  _E  ) )  i^i  ( Trans  X.  _V ) )  =  ( ( SSet 
i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) )
53, 4eqtri 2483 . . . . 5  |-  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ( ( SSet  i^i  ( Trans  X.  _V )
)  \  (  _I  u.  _E  ) )
65rneqi 5177 . . . 4  |-  ran  (
( SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ran  ( ( SSet  i^i  ( Trans  X.  _V )
)  \  (  _I  u.  _E  ) )
72, 6eqtri 2483 . . 3  |-  ( (
SSet  \  (  _I  u.  _E  ) ) " Trans )  =  ran  ( (
SSet  i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) )
87difeq2i 3582 . 2  |-  ( _V 
\  ( ( SSet  \  (  _I  u.  _E  ) ) " Trans ) )  =  ( _V 
\  ran  ( ( SSet  i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) ) )
91, 8eqtr4i 2486 1  |-  On  =  ( _V  \  (
( SSet  \  (  _I  u.  _E  ) )
" Trans ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370   _Vcvv 3078    \ cdif 3436    u. cun 3437    i^i cin 3438    _E cep 4741    _I cid 4742   Oncon0 4830    X. cxp 4949   ran crn 4952    |` cres 4953   "cima 4954   SSetcsset 28029   Transctrans 28030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fo 5535  df-fv 5537  df-1st 6690  df-2nd 6691  df-txp 28051  df-sset 28053  df-trans 28054
This theorem is referenced by: (None)
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