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Theorem subgruhgrfun 26394
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 2760 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
21uhgrfun 26181 . 2 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
3 subgrfun 26393 . 2 ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
42, 3sylan 489 1 ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139   class class class wbr 4804  Fun wfun 6043  cfv 6049  iEdgciedg 26095  UHGraphcuhgr 26171   SubGraph csubgr 26379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-uhgr 26173  df-subgr 26380
This theorem is referenced by:  subgruhgredgd  26396  subuhgr  26398  subupgr  26399  subumgr  26400  subusgr  26401
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