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Mirrors > Home > MPE Home > Th. List > 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 | edgval 26842 | . . 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 27313 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (iEdg‘𝐺) = {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}) |
5 | 4 | rneqd 5772 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran (iEdg‘𝐺) = ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}) |
6 | c0ex 10624 | . . . . 5 ⊢ 0 ∈ V | |
7 | 1ex 10626 | . . . . 5 ⊢ 1 ∈ V | |
8 | rnpropg 6046 | . . . . 5 ⊢ ((0 ∈ V ∧ 1 ∈ V) → ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉} = {{𝐴, 𝐵}, {𝐴, 𝐵}}) | |
9 | 6, 7, 8 | mp2an 691 | . . . 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 4538 | . . 3 ⊢ {{𝐴, 𝐵}} = {{𝐴, 𝐵}, {𝐴, 𝐵}} | |
12 | 10, 11 | eqtr4di 2851 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉} = {{𝐴, 𝐵}}) |
13 | 2, 5, 12 | 3eqtrd 2837 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 {cpr 4527 〈cop 4531 ran crn 5520 ‘cfv 6324 0cc0 10526 1c1 10527 iEdgciedg 26790 Edgcedg 26840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-mulcl 10588 ax-i2m1 10594 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-2nd 7672 df-iedg 26792 df-edg 26841 |
This theorem is referenced by: umgr2v2enb1 27316 |
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