![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > uhgriedg0edg0 | Structured version Visualization version GIF version |
Description: A hypergraph has no edges iff its edge function is empty. (Contributed by AV, 21-Oct-2020.) (Proof shortened by AV, 8-Dec-2021.) |
Ref | Expression |
---|---|
uhgriedg0edg0 | ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2825 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
2 | 1 | uhgrfun 26371 | . 2 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) |
3 | eqid 2825 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
4 | 1, 3 | edg0iedg0 26360 | . 2 ⊢ (Fun (iEdg‘𝐺) → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
5 | 2, 4 | syl 17 | 1 ⊢ (𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1656 ∈ wcel 2164 ∅c0 4146 Fun wfun 6121 ‘cfv 6127 iEdgciedg 26302 Edgcedg 26352 UHGraphcuhgr 26361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 df-edg 26353 df-uhgr 26363 |
This theorem is referenced by: uhgr0v0e 26542 uhgr0vusgr 26546 lfuhgr1v0e 26558 usgr1vr 26559 usgr1v0e 26630 uhgr0edg0rgr 26878 rgrusgrprc 26894 |
Copyright terms: Public domain | W3C validator |