Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  subgrfun Structured version   Visualization version   GIF version

Theorem subgrfun 27067
 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 2824 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
2 eqid 2824 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
3 eqid 2824 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
4 eqid 2824 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
5 eqid 2824 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 27060 . . 3 (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))
7 funss 6362 . . . 4 ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → (Fun (iEdg‘𝐺) → Fun (iEdg‘𝑆)))
873ad2ant2 1131 . . 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 411 1 ((Fun (iEdg‘𝐺) ∧ 𝑆 SubGraph 𝐺) → Fun (iEdg‘𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   ⊆ wss 3919  𝒫 cpw 4521   class class class wbr 5052  Fun wfun 6337  ‘cfv 6343  Vtxcvtx 26785  iEdgciedg 26786  Edgcedg 26836   SubGraph csubgr 27053 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-res 5554  df-iota 6302  df-fun 6345  df-fv 6351  df-subgr 27054 This theorem is referenced by:  subgruhgrfun  27068
 Copyright terms: Public domain W3C validator