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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 6667 | . 2 ⊢ (iEdg‘𝐺) = (iEdg‘〈𝑉, {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}〉) |
3 | simp1 1132 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → 𝑉 ∈ 𝑊) | |
4 | prex 5324 | . . 3 ⊢ {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉} ∈ V | |
5 | opiedgfv 26786 | . . 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 588 | . 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 2868 | 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 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {cpr 4562 〈cop 4566 ‘cfv 6349 0cc0 10531 1c1 10532 iEdgciedg 26776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fv 6357 df-2nd 7684 df-iedg 26778 |
This theorem is referenced by: umgr2v2eedg 27300 umgr2v2e 27301 umgr2v2evd2 27303 |
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