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Theorem subgruhgrfun 26991
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.)
Assertion
Ref Expression
subgruhgrfun ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))

Proof of Theorem subgruhgrfun
StepHypRef Expression
1 eqid 2818 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
21uhgrfun 26778 . 2 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
3 subgrfun 26990 . 2 ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
42, 3sylan 580 1 ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105   class class class wbr 5057  Fun wfun 6342  cfv 6348  iEdgciedg 26709  UHGraphcuhgr 26768   SubGraph csubgr 26976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-uhgr 26770  df-subgr 26977
This theorem is referenced by:  subgruhgredgd  26993  subuhgr  26995  subupgr  26996  subumgr  26997  subusgr  26998
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