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Theorem dfon4 24433
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 24432 . 2  |-  On  =  ( _V  \  ran  (
( SSet  i^i  ( Trans  X.  _V ) ) 
\  (  _I  u.  _E  ) ) )
2 df-ima 4702 . . . 4  |-  ( (
SSet  \  (  _I  u.  _E  ) ) " Trans )  =  ran  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )
3 df-res 4701 . . . . . 6  |-  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ( ( SSet  \  (  _I  u.  _E  ) )  i^i  ( Trans  X.  _V ) )
4 indif1 3413 . . . . . 6  |-  ( (
SSet  \  (  _I  u.  _E  ) )  i^i  ( Trans  X.  _V ) )  =  ( ( SSet 
i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) )
53, 4eqtri 2303 . . . . 5  |-  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ( ( SSet  i^i  ( Trans  X.  _V )
)  \  (  _I  u.  _E  ) )
65rneqi 4905 . . . 4  |-  ran  (
( SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ran  ( ( SSet  i^i  ( Trans  X.  _V )
)  \  (  _I  u.  _E  ) )
72, 6eqtri 2303 . . 3  |-  ( (
SSet  \  (  _I  u.  _E  ) ) " Trans )  =  ran  ( (
SSet  i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) )
87difeq2i 3291 . 2  |-  ( _V 
\  ( ( SSet  \  (  _I  u.  _E  ) ) " Trans ) )  =  ( _V 
\  ran  ( ( SSet  i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) ) )
91, 8eqtr4i 2306 1  |-  On  =  ( _V  \  (
( SSet  \  (  _I  u.  _E  ) )
" Trans ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    _E cep 4303    _I cid 4304   Oncon0 4392    X. cxp 4687   ran crn 4690    |` cres 4691   "cima 4692   SSetcsset 24375   Transctrans 24376
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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 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-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-txp 24395  df-sset 24397  df-trans 24398
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