Theorem dfon4 35854

Description: Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)

Assertion

Ref	Expression
dfon4	⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))

Proof of Theorem dfon4

Step	Hyp	Ref	Expression
1		dfon3 35853	. 2 ⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
2		df-ima 5644	. . . 4 ⊢ (( SSet ∖ ( I ∪ E )) “ Trans ) = ran (( SSet ∖ ( I ∪ E )) ↾ Trans )
3		df-res 5643	. . . . . 6 ⊢ (( SSet ∖ ( I ∪ E )) ↾ Trans ) = (( SSet ∖ ( I ∪ E )) ∩ ( Trans × V))
4		indif1 4241	. . . . . 6 ⊢ (( SSet ∖ ( I ∪ E )) ∩ ( Trans × V)) = (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
5	3, 4	eqtri 2752	. . . . 5 ⊢ (( SSet ∖ ( I ∪ E )) ↾ Trans ) = (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
6	5	rneqi 5890	. . . 4 ⊢ ran (( SSet ∖ ( I ∪ E )) ↾ Trans ) = ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
7	2, 6	eqtri 2752	. . 3 ⊢ (( SSet ∖ ( I ∪ E )) “ Trans ) = ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
8	7	difeq2i 4082	. 2 ⊢ (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans )) = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
9	1, 8	eqtr4i 2755	1 ⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))

Syntax hints:   = wceq 1540  Vcvv 3444   ∖ cdif 3908   ∪ cun 3909   ∩ cin 3910   I cid 5525   E cep 5530   × cxp 5629  ran crn 5632   ↾ cres 5633   “ cima 5634  Oncon0 6320   SSet csset 35793   Trans ctrans 35794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fo 6505  df-fv 6507  df-1st 7947  df-2nd 7948  df-txp 35815  df-sset 35817  df-trans 35818

This theorem is referenced by: (None)