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Theorem dfon4 30218
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 30217 . 2  |-  On  =  ( _V  \  ran  (
( SSet  i^i  ( Trans  X.  _V ) ) 
\  (  _I  u.  _E  ) ) )
2 df-ima 4835 . . . 4  |-  ( (
SSet  \  (  _I  u.  _E  ) ) " Trans )  =  ran  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )
3 df-res 4834 . . . . . 6  |-  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ( ( SSet  \  (  _I  u.  _E  ) )  i^i  ( Trans  X.  _V ) )
4 indif1 3693 . . . . . 6  |-  ( (
SSet  \  (  _I  u.  _E  ) )  i^i  ( Trans  X.  _V ) )  =  ( ( SSet 
i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) )
53, 4eqtri 2431 . . . . 5  |-  ( (
SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ( ( SSet  i^i  ( Trans  X.  _V )
)  \  (  _I  u.  _E  ) )
65rneqi 5049 . . . 4  |-  ran  (
( SSet  \  (  _I  u.  _E  ) )  |`  Trans )  =  ran  ( ( SSet  i^i  ( Trans  X.  _V )
)  \  (  _I  u.  _E  ) )
72, 6eqtri 2431 . . 3  |-  ( (
SSet  \  (  _I  u.  _E  ) ) " Trans )  =  ran  ( (
SSet  i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) )
87difeq2i 3557 . 2  |-  ( _V 
\  ( ( SSet  \  (  _I  u.  _E  ) ) " Trans ) )  =  ( _V 
\  ran  ( ( SSet  i^i  ( Trans  X.  _V ) )  \  (  _I  u.  _E  ) ) )
91, 8eqtr4i 2434 1  |-  On  =  ( _V  \  (
( SSet  \  (  _I  u.  _E  ) )
" Trans ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405   _Vcvv 3058    \ cdif 3410    u. cun 3411    i^i cin 3412    _E cep 4731    _I cid 4732    X. cxp 4820   ran crn 4823    |` cres 4824   "cima 4825   Oncon0 5409   SSetcsset 30156   Transctrans 30157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-ord 5412  df-on 5413  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fo 5574  df-fv 5576  df-1st 6783  df-2nd 6784  df-txp 30178  df-sset 30180  df-trans 30181
This theorem is referenced by: (None)
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