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Theorem subgruhgrfun 29300
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 2736 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
21uhgrfun 29084 . 2 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
3 subgrfun 29299 . 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 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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