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Mirrors > Home > MPE Home > Th. List > Mathboxes > umgr2v2eedg | Structured version Visualization version GIF version |
Description: The set of edges in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.) |
Ref | Expression |
---|---|
umgr2v2evtx.g | ⊢ 𝐺 = 〈𝑉, {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}〉 |
Ref | Expression |
---|---|
umgr2v2eedg | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgr2v2evtx.g | . . . 4 ⊢ 𝐺 = 〈𝑉, {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}〉 | |
2 | opex 4859 | . . . 4 ⊢ 〈𝑉, {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}〉 ∈ V | |
3 | 1, 2 | eqeltri 2684 | . . 3 ⊢ 𝐺 ∈ V |
4 | edgaval 25794 | . . 3 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺)) | |
5 | 3, 4 | mp1i 13 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
6 | 1 | umgr2v2eiedg 40739 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (iEdg‘𝐺) = {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}) |
7 | 6 | rneqd 5274 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran (iEdg‘𝐺) = ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}) |
8 | c0ex 9913 | . . . . 5 ⊢ 0 ∈ V | |
9 | 1ex 9914 | . . . . 5 ⊢ 1 ∈ V | |
10 | rnpropg 5533 | . . . . 5 ⊢ ((0 ∈ V ∧ 1 ∈ V) → ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉} = {{𝐴, 𝐵}, {𝐴, 𝐵}}) | |
11 | 8, 9, 10 | mp2an 704 | . . . 4 ⊢ ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉} = {{𝐴, 𝐵}, {𝐴, 𝐵}} |
12 | 11 | a1i 11 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉} = {{𝐴, 𝐵}, {𝐴, 𝐵}}) |
13 | dfsn2 4138 | . . 3 ⊢ {{𝐴, 𝐵}} = {{𝐴, 𝐵}, {𝐴, 𝐵}} | |
14 | 12, 13 | syl6eqr 2662 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉} = {{𝐴, 𝐵}}) |
15 | 5, 7, 14 | 3eqtrd 2648 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 {cpr 4127 〈cop 4131 ran crn 5039 ‘cfv 5804 0cc0 9815 1c1 9816 iEdgciedg 25674 Edgcedga 25792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-mulcl 9877 ax-i2m1 9883 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-2nd 7060 df-iedg 25676 df-edga 25793 |
This theorem is referenced by: umgr2v2enb1 40742 |
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