Theorem subgruhgrfun 28577

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

Step	Hyp	Ref	Expression
1		eqid 2732	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	uhgrfun 28364	. 2 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
3		subgrfun 28576	. 2 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
4	2, 3	sylan 580	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106   class class class wbr 5148  Fun wfun 6537  ‘cfv 6543  iEdgciedg 28295  UHGraphcuhgr 28354   SubGraph csubgr 28562

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-uhgr 28356  df-subgr 28563

This theorem is referenced by:  subgruhgredgd  28579  subuhgr  28581  subupgr  28582  subumgr  28583  subusgr  28584