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| 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 2736 | . . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
| 2 | 1 | uhgrfun 29084 | . 2 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺)) | 
| 3 | subgrfun 29299 | . 2 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆)) | |
| 4 | 2, 3 | sylan 580 | 1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 class class class wbr 5142 Fun wfun 6554 ‘cfv 6560 iEdgciedg 29015 UHGraphcuhgr 29074 SubGraph csubgr 29285 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-uhgr 29076 df-subgr 29286 | 
| This theorem is referenced by: subgruhgredgd 29302 subuhgr 29304 subupgr 29305 subumgr 29306 subusgr 29307 | 
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