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Theorem umgr2v2eedg 26654
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 26161 . . 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 26653 . . 3 ((𝑉𝑊𝐴𝑉𝐵𝑉) → (iEdg‘𝐺) = {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
54rneqd 5491 . 2 ((𝑉𝑊𝐴𝑉𝐵𝑉) → ran (iEdg‘𝐺) = ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
6 c0ex 10235 . . . . 5 0 ∈ V
7 1ex 10236 . . . . 5 1 ∈ V
8 rnpropg 5757 . . . . 5 ((0 ∈ V ∧ 1 ∈ V) → ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}, {𝐴, 𝐵}})
96, 7, 8mp2an 664 . . . 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 4329 . . 3 {{𝐴, 𝐵}} = {{𝐴, 𝐵}, {𝐴, 𝐵}}
1210, 11syl6eqr 2823 . 2 ((𝑉𝑊𝐴𝑉𝐵𝑉) → ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}})
132, 5, 123eqtrd 2809 1 ((𝑉𝑊𝐴𝑉𝐵𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  {csn 4316  {cpr 4318  cop 4322  ran crn 5250  cfv 6031  0cc0 10137  1c1 10138  iEdgciedg 26095  Edgcedg 26159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7095  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-mulcl 10199  ax-i2m1 10205
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-2nd 7315  df-iedg 26097  df-edg 26160
This theorem is referenced by:  umgr2v2enb1  26656
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