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Theorem umgr2v2eedg 26651
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 26650 . . 3 ((𝑉𝑊𝐴𝑉𝐵𝑉) → (iEdg‘𝐺) = {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
54rneqd 5508 . 2 ((𝑉𝑊𝐴𝑉𝐵𝑉) → ran (iEdg‘𝐺) = ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
6 c0ex 10246 . . . . 5 0 ∈ V
7 1ex 10247 . . . . 5 1 ∈ V
8 rnpropg 5774 . . . . 5 ((0 ∈ V ∧ 1 ∈ V) → ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}, {𝐴, 𝐵}})
96, 7, 8mp2an 710 . . . 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 4334 . . 3 {{𝐴, 𝐵}} = {{𝐴, 𝐵}, {𝐴, 𝐵}}
1210, 11syl6eqr 2812 . 2 ((𝑉𝑊𝐴𝑉𝐵𝑉) → ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}})
132, 5, 123eqtrd 2798 1 ((𝑉𝑊𝐴𝑉𝐵𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  wcel 2139  Vcvv 3340  {csn 4321  {cpr 4323  cop 4327  ran crn 5267  cfv 6049  0cc0 10148  1c1 10149  iEdgciedg 26095  Edgcedg 26159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-mulcl 10210  ax-i2m1 10216
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-2nd 7335  df-iedg 26097  df-edg 26160
This theorem is referenced by:  umgr2v2enb1  26653
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