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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 26417 | . 2 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉) |
3 | eqcom 2629 | . . . . 5 ⊢ ((Vtx‘𝐺) = 𝑉 ↔ 𝑉 = (Vtx‘𝐺)) | |
4 | 3 | biimpi 206 | . . . 4 ⊢ ((Vtx‘𝐺) = 𝑉 → 𝑉 = (Vtx‘𝐺)) |
5 | 4 | eleq2d 2687 | . . 3 ⊢ ((Vtx‘𝐺) = 𝑉 → (𝐴 ∈ 𝑉 ↔ 𝐴 ∈ (Vtx‘𝐺))) |
6 | 5 | biimpcd 239 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((Vtx‘𝐺) = 𝑉 → 𝐴 ∈ (Vtx‘𝐺))) |
7 | 2, 6 | mpan9 486 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ (Vtx‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {cpr 4179 〈cop 4183 ‘cfv 5888 0cc0 9936 1c1 9937 Vtxcvtx 25874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-1st 7168 df-vtx 25876 |
This theorem is referenced by: umgr2v2enb1 26422 umgr2v2evd2 26423 |
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