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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 29066 | . . 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 29541 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (iEdg‘𝐺) = {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}) |
| 5 | 4 | rneqd 5949 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran (iEdg‘𝐺) = ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}) |
| 6 | c0ex 11255 | . . . . 5 ⊢ 0 ∈ V | |
| 7 | 1ex 11257 | . . . . 5 ⊢ 1 ∈ V | |
| 8 | rnpropg 6242 | . . . . 5 ⊢ ((0 ∈ V ∧ 1 ∈ V) → ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉} = {{𝐴, 𝐵}, {𝐴, 𝐵}}) | |
| 9 | 6, 7, 8 | mp2an 692 | . . . 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 4639 | . . 3 ⊢ {{𝐴, 𝐵}} = {{𝐴, 𝐵}, {𝐴, 𝐵}} | |
| 12 | 10, 11 | eqtr4di 2795 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉} = {{𝐴, 𝐵}}) |
| 13 | 2, 5, 12 | 3eqtrd 2781 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 {cpr 4628 〈cop 4632 ran crn 5686 ‘cfv 6561 0cc0 11155 1c1 11156 iEdgciedg 29014 Edgcedg 29064 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-i2m1 11223 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-2nd 8015 df-iedg 29016 df-edg 29065 |
| This theorem is referenced by: umgr2v2enb1 29544 |
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