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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 2622 | . . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | uhgrfun.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | uhgrf 25957 | . 2 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
4 | 3 | ffund 6049 | 1 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ∅c0 3915 𝒫 cpw 4158 {csn 4177 dom cdm 5114 Fun wfun 5882 ‘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-nul 4789 |
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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 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-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 upgrle2 26000 uhgredgiedgb 26021 uhgriedg0edg0 26022 uhgrvtxedgiedgb 26031 edglnl 26038 numedglnl 26039 uhgr2edg 26100 ushgredgedg 26121 ushgredgedgloop 26123 0uhgrsubgr 26171 uhgrsubgrself 26172 subgruhgrfun 26174 subgruhgredgd 26176 subumgredg2 26177 subupgr 26179 uhgrspansubgrlem 26182 uhgrspansubgr 26183 uhgrspan1 26195 upgrreslem 26196 umgrreslem 26197 upgrres 26198 umgrres 26199 vtxduhgr0e 26374 vtxduhgrun 26379 vtxduhgrfiun 26380 finsumvtxdg2ssteplem1 26441 upgrewlkle2 26502 upgredginwlk 26532 wlkiswwlks1 26753 wlkiswwlksupgr2 26763 umgrwwlks2on 26850 vdn0conngrumgrv2 27056 eulerpathpr 27100 eulercrct 27102 |
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