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Mirrors > Home > MPE Home > Th. List > vtxdgop | Structured version Visualization version Unicode version |
Description: The vertex degree expressed as operation. (Contributed by AV, 12-Dec-2021.) |
Ref | Expression |
---|---|
vtxdgop | VtxDeg VtxVtxDegiEdg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 4932 | . . 3 Vtx iEdg | |
2 | fvex 6201 | . . . . . 6 Vtx | |
3 | fvex 6201 | . . . . . 6 iEdg | |
4 | 2, 3 | opvtxfvi 25889 | . . . . 5 VtxVtx iEdg Vtx |
5 | 4 | eqcomi 2631 | . . . 4 Vtx VtxVtx iEdg |
6 | 2, 3 | opiedgfvi 25890 | . . . . 5 iEdgVtx iEdg iEdg |
7 | 6 | eqcomi 2631 | . . . 4 iEdg iEdgVtx iEdg |
8 | eqid 2622 | . . . 4 iEdg iEdg | |
9 | 5, 7, 8 | vtxdgfval 26363 | . . 3 Vtx iEdg VtxDegVtx iEdg Vtx iEdg iEdg iEdg iEdg |
10 | 1, 9 | mp1i 13 | . 2 VtxDegVtx iEdg Vtx iEdg iEdg iEdg iEdg |
11 | df-ov 6653 | . . 3 VtxVtxDegiEdg VtxDegVtx iEdg | |
12 | 11 | a1i 11 | . 2 VtxVtxDegiEdg VtxDegVtx iEdg |
13 | eqid 2622 | . . 3 Vtx Vtx | |
14 | eqid 2622 | . . 3 iEdg iEdg | |
15 | 13, 14, 8 | vtxdgfval 26363 | . 2 VtxDeg Vtx iEdg iEdg iEdg iEdg |
16 | 10, 12, 15 | 3eqtr4rd 2667 | 1 VtxDeg VtxVtxDegiEdg |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 crab 2916 cvv 3200 csn 4177 cop 4183 cmpt 4729 cdm 5114 cfv 5888 (class class class)co 6650 cxad 11944 chash 13117 Vtxcvtx 25874 iEdgciedg 25875 VtxDegcvtxdg 26361 |
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-rep 4771 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-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-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-1st 7168 df-2nd 7169 df-vtx 25876 df-iedg 25877 df-vtxdg 26362 |
This theorem is referenced by: finsumvtxdg2size 26446 |
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