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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 31169 | . 2 ⊢ On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) | |
2 | df-ima 5051 | . . . 4 ⊢ (( SSet ∖ ( I ∪ E )) “ Trans ) = ran (( SSet ∖ ( I ∪ E )) ↾ Trans ) | |
3 | df-res 5050 | . . . . . 6 ⊢ (( SSet ∖ ( I ∪ E )) ↾ Trans ) = (( SSet ∖ ( I ∪ E )) ∩ ( Trans × V)) | |
4 | indif1 3830 | . . . . . 6 ⊢ (( SSet ∖ ( I ∪ E )) ∩ ( Trans × V)) = (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) | |
5 | 3, 4 | eqtri 2632 | . . . . 5 ⊢ (( SSet ∖ ( I ∪ E )) ↾ Trans ) = (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) |
6 | 5 | rneqi 5273 | . . . 4 ⊢ ran (( SSet ∖ ( I ∪ E )) ↾ Trans ) = ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) |
7 | 2, 6 | eqtri 2632 | . . 3 ⊢ (( SSet ∖ ( I ∪ E )) “ Trans ) = ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )) |
8 | 7 | difeq2i 3687 | . 2 ⊢ (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans )) = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))) |
9 | 1, 8 | eqtr4i 2635 | 1 ⊢ On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans )) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 E cep 4947 I cid 4948 × cxp 5036 ran crn 5039 ↾ cres 5040 “ cima 5041 Oncon0 5640 SSet csset 31108 Trans ctrans 31109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 df-2nd 7060 df-txp 31130 df-sset 31132 df-trans 31133 |
This theorem is referenced by: (None) |
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