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Mirrors > Home > MPE Home > Th. List > subgruhgrfun | Structured version Visualization version GIF version |
Description: The edge function of a subgraph of a hypergraph is a function. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 20-Nov-2020.) |
Ref | Expression |
---|---|
subgruhgrfun | ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
2 | 1 | uhgrfun 26859 | . 2 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) |
3 | subgrfun 27071 | . 2 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆)) | |
4 | 2, 3 | sylan 583 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 class class class wbr 5030 Fun wfun 6318 ‘cfv 6324 iEdgciedg 26790 UHGraphcuhgr 26849 SubGraph csubgr 27057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-uhgr 26851 df-subgr 27058 |
This theorem is referenced by: subgruhgredgd 27074 subuhgr 27076 subupgr 27077 subumgr 27078 subusgr 27079 |
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