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| Mirrors > Home > MPE Home > Th. List > uhgrfun | Structured version Visualization version GIF version | ||
| Description: The edge function of an undirected hypergraph is a function. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 15-Dec-2020.) |
| Ref | Expression |
|---|---|
| uhgrfun.e | ⊢ 𝐸 = (iEdg‘𝐺) |
| Ref | Expression |
|---|---|
| uhgrfun | ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
| 2 | uhgrfun.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
| 3 | 1, 2 | uhgrf 29079 | . 2 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
| 4 | 3 | ffund 6740 | 1 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∅c0 4333 𝒫 cpw 4600 {csn 4626 dom cdm 5685 Fun wfun 6555 ‘cfv 6561 Vtxcvtx 29013 iEdgciedg 29014 UHGraphcuhgr 29073 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-uhgr 29075 |
| This theorem is referenced by: lpvtx 29085 upgrle2 29122 uhgredgiedgb 29143 uhgriedg0edg0 29144 uhgrvtxedgiedgb 29153 edglnl 29160 numedglnl 29161 uhgr2edg 29225 ushgredgedg 29246 ushgredgedgloop 29248 0uhgrsubgr 29296 uhgrsubgrself 29297 subgruhgrfun 29299 subgruhgredgd 29301 subumgredg2 29302 subupgr 29304 uhgrspansubgrlem 29307 uhgrspansubgr 29308 uhgrspan1 29320 upgrreslem 29321 umgrreslem 29322 upgrres 29323 umgrres 29324 vtxduhgr0e 29496 vtxduhgrun 29501 vtxduhgrfiun 29502 finsumvtxdg2ssteplem1 29563 upgrewlkle2 29624 upgredginwlk 29654 wlkiswwlks1 29887 wlkiswwlksupgr2 29897 umgrwwlks2on 29977 vdn0conngrumgrv2 30215 eulerpathpr 30259 eulercrct 30261 lfuhgr 35123 loop1cycl 35142 umgr2cycllem 35145 isubgrvtxuhgr 47850 isubgredg 47852 isubgrsubgr 47855 isubgr0uhgr 47859 isuspgrim0lem 47871 isuspgrim0 47872 clnbgrgrimlem 47901 clnbgrgrim 47902 grimedg 47903 |
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