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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 2727 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
2 | 1 | uhgrfun 28866 | . 2 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) |
3 | eqid 2727 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺) | |
4 | 1, 3 | edg0iedg0 28855 | . 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 205 = wceq 1534 ∈ wcel 2099 ∅c0 4318 Fun wfun 6536 ‘cfv 6542 iEdgciedg 28797 Edgcedg 28847 UHGraphcuhgr 28856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-edg 28848 df-uhgr 28858 |
This theorem is referenced by: uhgr0v0e 29038 uhgr0vusgr 29042 lfuhgr1v0e 29054 usgr1vr 29055 usgr1v0e 29126 uhgr0edg0rgr 29374 rgrusgrprc 29390 |
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