Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > dfon4 | Structured version Visualization version GIF version |
Description: Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.) |
Ref | Expression |
---|---|
dfon4 | ⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfon3 31999 | . 2 ⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) | |
2 | df-ima 5127 | . . . 4 ⊢ (( SSet ∖ ( I ∪ E )) “ Trans ) = ran (( SSet ∖ ( I ∪ E )) ↾ Trans ) | |
3 | df-res 5126 | . . . . . 6 ⊢ (( SSet ∖ ( I ∪ E )) ↾ Trans ) = (( SSet ∖ ( I ∪ E )) ∩ ( Trans × V)) | |
4 | indif1 3871 | . . . . . 6 ⊢ (( SSet ∖ ( I ∪ E )) ∩ ( Trans × V)) = (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) | |
5 | 3, 4 | eqtri 2644 | . . . . 5 ⊢ (( SSet ∖ ( I ∪ E )) ↾ Trans ) = (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) |
6 | 5 | rneqi 5352 | . . . 4 ⊢ ran (( SSet ∖ ( I ∪ E )) ↾ Trans ) = ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) |
7 | 2, 6 | eqtri 2644 | . . 3 ⊢ (( SSet ∖ ( I ∪ E )) “ Trans ) = ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) |
8 | 7 | difeq2i 3725 | . 2 ⊢ (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans )) = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) |
9 | 1, 8 | eqtr4i 2647 | 1 ⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans )) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 I cid 5023 E cep 5028 × cxp 5112 ran crn 5115 ↾ cres 5116 “ cima 5117 Oncon0 5723 SSet csset 31939 Trans ctrans 31940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-1st 7168 df-2nd 7169 df-txp 31961 df-sset 31963 df-trans 31964 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |