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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 29081 | . . 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 29556 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (iEdg‘𝐺) = {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}) |
5 | 4 | rneqd 5952 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran (iEdg‘𝐺) = ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}) |
6 | c0ex 11253 | . . . . 5 ⊢ 0 ∈ V | |
7 | 1ex 11255 | . . . . 5 ⊢ 1 ∈ V | |
8 | rnpropg 6244 | . . . . 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 4644 | . . 3 ⊢ {{𝐴, 𝐵}} = {{𝐴, 𝐵}, {𝐴, 𝐵}} | |
12 | 10, 11 | eqtr4di 2793 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ran {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉} = {{𝐴, 𝐵}}) |
13 | 2, 5, 12 | 3eqtrd 2779 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 {cpr 4633 〈cop 4637 ran crn 5690 ‘cfv 6563 0cc0 11153 1c1 11154 iEdgciedg 29029 Edgcedg 29079 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-i2m1 11221 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-2nd 8014 df-iedg 29031 df-edg 29080 |
This theorem is referenced by: umgr2v2enb1 29559 |
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