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Mirrors > Home > MPE Home > Th. List > uhgredgiedgb | Structured version Visualization version GIF version |
Description: In a hypergraph, a set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.) |
Ref | Expression |
---|---|
uhgredgiedgb.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
uhgredgiedgb | ⊢ (𝐺 ∈ UHGraph → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgredgiedgb.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
2 | 1 | uhgrfun 26854 | . 2 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼) |
3 | 1 | edgiedgb 26842 | . 2 ⊢ (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥))) |
4 | 2, 3 | syl 17 | 1 ⊢ (𝐺 ∈ UHGraph → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 dom cdm 5558 Fun wfun 6352 ‘cfv 6358 iEdgciedg 26785 Edgcedg 26835 UHGraphcuhgr 26844 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-edg 26836 df-uhgr 26846 |
This theorem is referenced by: usgredg2vtxeuALT 27007 vtxduhgr0nedg 27277 umgr2wlk 27731 1pthon2v 27935 uhgr3cyclex 27964 |
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