MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  umgr2v2eedg Structured version   Visualization version   GIF version

Theorem umgr2v2eedg 29542
Description: The set of edges in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
Hypothesis
Ref Expression
umgr2v2evtx.g 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩
Assertion
Ref Expression
umgr2v2eedg ((𝑉𝑊𝐴𝑉𝐵𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}})

Proof of Theorem umgr2v2eedg
StepHypRef Expression
1 edgval 29066 . . 3 (Edg‘𝐺) = ran (iEdg‘𝐺)
21a1i 11 . 2 ((𝑉𝑊𝐴𝑉𝐵𝑉) → (Edg‘𝐺) = ran (iEdg‘𝐺))
3 umgr2v2evtx.g . . . 4 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩
43umgr2v2eiedg 29541 . . 3 ((𝑉𝑊𝐴𝑉𝐵𝑉) → (iEdg‘𝐺) = {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
54rneqd 5949 . 2 ((𝑉𝑊𝐴𝑉𝐵𝑉) → ran (iEdg‘𝐺) = ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
6 c0ex 11255 . . . . 5 0 ∈ V
7 1ex 11257 . . . . 5 1 ∈ V
8 rnpropg 6242 . . . . 5 ((0 ∈ V ∧ 1 ∈ V) → ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}, {𝐴, 𝐵}})
96, 7, 8mp2an 692 . . . 4 ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}, {𝐴, 𝐵}}
109a1i 11 . . 3 ((𝑉𝑊𝐴𝑉𝐵𝑉) → ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}, {𝐴, 𝐵}})
11 dfsn2 4639 . . 3 {{𝐴, 𝐵}} = {{𝐴, 𝐵}, {𝐴, 𝐵}}
1210, 11eqtr4di 2795 . 2 ((𝑉𝑊𝐴𝑉𝐵𝑉) → ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}})
132, 5, 123eqtrd 2781 1 ((𝑉𝑊𝐴𝑉𝐵𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  {cpr 4628  cop 4632  ran crn 5686  cfv 6561  0cc0 11155  1c1 11156  iEdgciedg 29014  Edgcedg 29064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-mulcl 11217  ax-i2m1 11223
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-2nd 8015  df-iedg 29016  df-edg 29065
This theorem is referenced by:  umgr2v2enb1  29544
  Copyright terms: Public domain W3C validator