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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 26834 | . . 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 27305 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (iEdg‘𝐺) = {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}) |
5 | 4 | rneqd 5808 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran (iEdg‘𝐺) = ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}) |
6 | c0ex 10635 | . . . . 5 ⊢ 0 ∈ V | |
7 | 1ex 10637 | . . . . 5 ⊢ 1 ∈ V | |
8 | rnpropg 6079 | . . . . 5 ⊢ ((0 ∈ V ∧ 1 ∈ V) → ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉} = {{𝐴, 𝐵}, {𝐴, 𝐵}}) | |
9 | 6, 7, 8 | mp2an 690 | . . . 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 4580 | . . 3 ⊢ {{𝐴, 𝐵}} = {{𝐴, 𝐵}, {𝐴, 𝐵}} | |
12 | 10, 11 | syl6eqr 2874 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉} = {{𝐴, 𝐵}}) |
13 | 2, 5, 12 | 3eqtrd 2860 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 {cpr 4569 〈cop 4573 ran crn 5556 ‘cfv 6355 0cc0 10537 1c1 10538 iEdgciedg 26782 Edgcedg 26832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-mulcl 10599 ax-i2m1 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fv 6363 df-2nd 7690 df-iedg 26784 df-edg 26833 |
This theorem is referenced by: umgr2v2enb1 27308 |
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