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Theorem umgr2v2eedg 26300
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 umgr2v2evtx.g . . . 4 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩
2 opex 4898 . . . 4 𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩ ∈ V
31, 2eqeltri 2700 . . 3 𝐺 ∈ V
4 edgval 25836 . . 3 (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺))
53, 4mp1i 13 . 2 ((𝑉𝑊𝐴𝑉𝐵𝑉) → (Edg‘𝐺) = ran (iEdg‘𝐺))
61umgr2v2eiedg 26299 . . 3 ((𝑉𝑊𝐴𝑉𝐵𝑉) → (iEdg‘𝐺) = {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
76rneqd 5317 . 2 ((𝑉𝑊𝐴𝑉𝐵𝑉) → ran (iEdg‘𝐺) = ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
8 c0ex 9979 . . . . 5 0 ∈ V
9 1ex 9980 . . . . 5 1 ∈ V
10 rnpropg 5577 . . . . 5 ((0 ∈ V ∧ 1 ∈ V) → ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}, {𝐴, 𝐵}})
118, 9, 10mp2an 707 . . . 4 ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}, {𝐴, 𝐵}}
1211a1i 11 . . 3 ((𝑉𝑊𝐴𝑉𝐵𝑉) → ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}, {𝐴, 𝐵}})
13 dfsn2 4166 . . 3 {{𝐴, 𝐵}} = {{𝐴, 𝐵}, {𝐴, 𝐵}}
1412, 13syl6eqr 2678 . 2 ((𝑉𝑊𝐴𝑉𝐵𝑉) → ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}})
155, 7, 143eqtrd 2664 1 ((𝑉𝑊𝐴𝑉𝐵𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1992  Vcvv 3191  {csn 4153  {cpr 4155  cop 4159  ran crn 5080  cfv 5850  0cc0 9881  1c1 9882  iEdgciedg 25770  Edgcedg 25834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-mulcl 9943  ax-i2m1 9949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fv 5858  df-2nd 7117  df-iedg 25772  df-edg 25835
This theorem is referenced by:  umgr2v2enb1  26302
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