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Mirrors > Home > MPE Home > Th. List > uhgrn0 | Structured version Visualization version Unicode version |
Description: An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 15-Dec-2020.) |
Ref | Expression |
---|---|
uhgrfun.e | iEdg |
Ref | Expression |
---|---|
uhgrn0 | UHGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . . . 7 Vtx Vtx | |
2 | uhgrfun.e | . . . . . . 7 iEdg | |
3 | 1, 2 | uhgrf 25957 | . . . . . 6 UHGraph Vtx |
4 | fndm 5990 | . . . . . . 7 | |
5 | 4 | feq2d 6031 | . . . . . 6 Vtx Vtx |
6 | 3, 5 | syl5ibcom 235 | . . . . 5 UHGraph Vtx |
7 | 6 | imp 445 | . . . 4 UHGraph Vtx |
8 | 7 | ffvelrnda 6359 | . . 3 UHGraph Vtx |
9 | 8 | 3impa 1259 | . 2 UHGraph Vtx |
10 | eldifsni 4320 | . 2 Vtx | |
11 | 9, 10 | syl 17 | 1 UHGraph |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 cdif 3571 c0 3915 cpw 4158 csn 4177 cdm 5114 wfn 5883 wf 5884 cfv 5888 Vtxcvtx 25874 iEdgciedg 25875 UHGraph cuhgr 25951 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
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-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-uhgr 25953 |
This theorem is referenced by: lpvtx 25963 subgruhgredgd 26176 |
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