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Mirrors > Home > MPE Home > Th. List > umgr2v2eiedg | Structured version Visualization version GIF version |
Description: The edge function 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 |
---|---|
umgr2v2eiedg | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (iEdg‘𝐺) = {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgr2v2evtx.g | . . 3 ⊢ 𝐺 = 〈𝑉, {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}〉 | |
2 | 1 | fveq2i 6666 | . 2 ⊢ (iEdg‘𝐺) = (iEdg‘〈𝑉, {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}〉) |
3 | simp1 1128 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑉 ∈ 𝑊) | |
4 | prex 5323 | . . 3 ⊢ {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉} ∈ V | |
5 | opiedgfv 26719 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉} ∈ V) → (iEdg‘〈𝑉, {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}〉) = {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}) | |
6 | 3, 4, 5 | sylancl 586 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (iEdg‘〈𝑉, {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}〉) = {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}) |
7 | 2, 6 | syl5eq 2865 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (iEdg‘𝐺) = {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {cpr 4559 〈cop 4563 ‘cfv 6348 0cc0 10525 1c1 10526 iEdgciedg 26709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-2nd 7679 df-iedg 26711 |
This theorem is referenced by: umgr2v2eedg 27233 umgr2v2e 27234 umgr2v2evd2 27236 |
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