Theorem subgruhgrfun 29728

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 2762	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	uhgrfun 29509	. 2 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
3		subgrfun 29727	. 2 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
4	2, 3	sylan 592	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145   class class class wbr 5107  Fun wfun 6531  ‘cfv 6537  iEdgciedg 29440  UHGraphcuhgr 29499   SubGraph csubgr 29713

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-uhgr 29501  df-subgr 29714

This theorem is used by:  subgruhgredgd  29730  subuhgr  29732  subupgr  29733  subumgr  29734  subusgr  29735