Theorem subgrfun 26515

Description: The edge function of a subgraph of a graph whose edge function is actually a function is a function. (Contributed by AV, 20-Nov-2020.)

Assertion

Ref	Expression
subgrfun	⊢ ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))

Proof of Theorem subgrfun

Step	Hyp	Ref	Expression
1		eqid 2799	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
2		eqid 2799	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2799	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
4		eqid 2799	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		eqid 2799	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
6	1, 2, 3, 4, 5	subgrprop2 26508	. . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7		funss 6120	. . . 4 ⊢ ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → (Fun (iEdg‘𝐺) → Fun (iEdg‘𝑆)))
8	7	3ad2ant2 1165	. . 3 ⊢ (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → (Fun (iEdg‘𝐺) → Fun (iEdg‘𝑆)))
9	6, 8	syl 17	. 2 ⊢ (𝑆 SubGraph 𝐺 → (Fun (iEdg‘𝐺) → Fun (iEdg‘𝑆)))
10	9	impcom 397	1 ⊢ ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))

Syntax hints:   → wi 4   ∧ wa 385   ∧ w3a 1108   ⊆ wss 3769  𝒫 cpw 4349   class class class wbr 4843  Fun wfun 6095  ‘cfv 6101  Vtxcvtx 26231  iEdgciedg 26232  Edgcedg 26282   SubGraph csubgr 26501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-res 5324  df-iota 6064  df-fun 6103  df-fv 6109  df-subgr 26502

This theorem is referenced by:  subgruhgrfun  26516