Theorem subgruhgrfun 29598

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 2761	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
2	1	uhgrfun 29382	. 2 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
3		subgrfun 29597	. 2 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
4	2, 3	sylan 591	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2141   class class class wbr 5108  Fun wfun 6530  ‘cfv 6536  iEdgciedg 29313  UHGraphcuhgr 29372   SubGraph csubgr 29583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-uhgr 29374  df-subgr 29584

This theorem is referenced by:  subgruhgredgd  29600  subuhgr  29602  subupgr  29603  subumgr  29604  subusgr  29605