Theorem umgr2v2eedg 29618

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	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}})

Proof of Theorem umgr2v2eedg

Step	Hyp	Ref	Expression
1		edgval 29143	. . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
2	1	a1i 11	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (Edg‘𝐺) = ran (iEdg‘𝐺))
3		umgr2v2evtx.g	. . . 4 ⊢ 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩
4	3	umgr2v2eiedg 29617	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (iEdg‘𝐺) = {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
5	4	rneqd 5887	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran (iEdg‘𝐺) = ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
6		c0ex 11136	. . . . 5 ⊢ 0 ∈ V
7		1ex 11138	. . . . 5 ⊢ 1 ∈ V
8		rnpropg 6180	. . . . 5 ⊢ ((0 ∈ V ∧ 1 ∈ V) → ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}, {𝐴, 𝐵}})
9	6, 7, 8	mp2an 698	. . . 4 ⊢ ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}, {𝐴, 𝐵}}
10	9	a1i 11	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}, {𝐴, 𝐵}})
11		dfsn2 4575	. . 3 ⊢ {{𝐴, 𝐵}} = {{𝐴, 𝐵}, {𝐴, 𝐵}}
12	10, 11	eqtr4di 2793	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}})
13	2, 5, 12	3eqtrd 2779	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}})

Syntax hints:   → wi 4   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  Vcvv 3432  {csn 4562  {cpr 4564  ⟨cop 4568  ran crn 5626  ‘cfv 6492  0cc0 11036  1c1 11037  iEdgciedg 29091  Edgcedg 29141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-mulcl 11098  ax-i2m1 11104

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-2nd 7939  df-iedg 29093  df-edg 29142

This theorem is referenced by:  umgr2v2enb1  29620