Theorem dfon4 34291

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 34290	. 2 ⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
2		df-ima 5633	. . . 4 ⊢ (( SSet ∖ ( I ∪ E )) “ Trans ) = ran (( SSet ∖ ( I ∪ E )) ↾ Trans )
3		df-res 5632	. . . . . 6 ⊢ (( SSet ∖ ( I ∪ E )) ↾ Trans ) = (( SSet ∖ ( I ∪ E )) ∩ ( Trans × V))
4		indif1 4218	. . . . . 6 ⊢ (( SSet ∖ ( I ∪ E )) ∩ ( Trans × V)) = (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
5	3, 4	eqtri 2764	. . . . 5 ⊢ (( SSet ∖ ( I ∪ E )) ↾ Trans ) = (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
6	5	rneqi 5878	. . . 4 ⊢ ran (( SSet ∖ ( I ∪ E )) ↾ Trans ) = ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
7	2, 6	eqtri 2764	. . 3 ⊢ (( SSet ∖ ( I ∪ E )) “ Trans ) = ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
8	7	difeq2i 4066	. 2 ⊢ (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans )) = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
9	1, 8	eqtr4i 2767	1 ⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))

Syntax hints:   = wceq 1540  Vcvv 3441   ∖ cdif 3895   ∪ cun 3896   ∩ cin 3897   I cid 5517   E cep 5523   × cxp 5618  ran crn 5621   ↾ cres 5622   “ cima 5623  Oncon0 6302   SSet csset 34230   Trans ctrans 34231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-ord 6305  df-on 6306  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-fo 6485  df-fv 6487  df-1st 7899  df-2nd 7900  df-txp 34252  df-sset 34254  df-trans 34255

This theorem is referenced by: (None)