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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
StepHypRef 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)
61, 2, 3, 4, 5subgrprop2 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)))
873ad2ant2 1165 . . 3 (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → (Fun (iEdg‘𝐺) → Fun (iEdg‘𝑆)))
96, 8syl 17 . 2 (𝑆 SubGraph 𝐺 → (Fun (iEdg‘𝐺) → Fun (iEdg‘𝑆)))
109impcom 397 1 ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
Colors of variables: wff setvar class
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
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