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

Proof of Theorem umgr2v2eedg

Step	Hyp	Ref	Expression
1		edgval 26161	. . 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 26650	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (iEdg‘𝐺) = {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})
5	4	rneqd 5508	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 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}})
9	6, 7, 8	mp2an 710	. . . 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 4334	. . 3 ⊢ {{𝐴, 𝐵}} = {{𝐴, 𝐵}, {𝐴, 𝐵}}
12	10, 11	syl6eqr 2812	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} = {{𝐴, 𝐵}})
13	2, 5, 12	3eqtrd 2798	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}})

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