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Theorem umgr2v2eiedg 29376
Description: The edge function 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
umgr2v2eiedg ((𝑉𝑊𝐴𝑉𝐵𝑉) → (iEdg‘𝐺) = {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})

Proof of Theorem umgr2v2eiedg
StepHypRef Expression
1 umgr2v2evtx.g . . 3 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩
21fveq2i 6893 . 2 (iEdg‘𝐺) = (iEdg‘⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩)
3 simp1 1133 . . 3 ((𝑉𝑊𝐴𝑉𝐵𝑉) → 𝑉𝑊)
4 prex 5429 . . 3 {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} ∈ V
5 opiedgfv 28859 . . 3 ((𝑉𝑊 ∧ {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} ∈ V) → (iEdg‘⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩) = {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
63, 4, 5sylancl 584 . 2 ((𝑉𝑊𝐴𝑉𝐵𝑉) → (iEdg‘⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩) = {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
72, 6eqtrid 2777 1 ((𝑉𝑊𝐴𝑉𝐵𝑉) → (iEdg‘𝐺) = {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3463  {cpr 4627  cop 4631  cfv 6543  0cc0 11133  1c1 11134  iEdgciedg 28849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-iota 6495  df-fun 6545  df-fv 6551  df-2nd 7988  df-iedg 28851
This theorem is referenced by:  umgr2v2eedg  29377  umgr2v2e  29378  umgr2v2evd2  29380
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