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Mirrors > Home > MPE Home > Th. List > umgr2v2evtxel | Structured version Visualization version GIF version |
Description: A vertex 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 |
---|---|
umgr2v2evtxel | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgr2v2evtx.g | . . 3 ⊢ 𝐺 = 〈𝑉, {〈0, {𝐴, 𝐵}〉, 〈1, {𝐴, 𝐵}〉}〉 | |
2 | 1 | umgr2v2evtx 28298 | . 2 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉) |
3 | eqcom 2745 | . . . . 5 ⊢ ((Vtx‘𝐺) = 𝑉 ↔ 𝑉 = (Vtx‘𝐺)) | |
4 | 3 | biimpi 215 | . . . 4 ⊢ ((Vtx‘𝐺) = 𝑉 → 𝑉 = (Vtx‘𝐺)) |
5 | 4 | eleq2d 2824 | . . 3 ⊢ ((Vtx‘𝐺) = 𝑉 → (𝐴 ∈ 𝑉 ↔ 𝐴 ∈ (Vtx‘𝐺))) |
6 | 5 | biimpcd 249 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((Vtx‘𝐺) = 𝑉 → 𝐴 ∈ (Vtx‘𝐺))) |
7 | 2, 6 | mpan9 508 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cpr 4587 〈cop 4591 ‘cfv 6494 0cc0 11010 1c1 11011 Vtxcvtx 27776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 ax-un 7665 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-iota 6446 df-fun 6496 df-fv 6502 df-1st 7914 df-vtx 27778 |
This theorem is referenced by: umgr2v2enb1 28303 umgr2v2evd2 28304 |
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