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Mirrors > Home > MPE Home > Th. List > uspgrloopedg | Structured version Visualization version GIF version |
Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 27039) is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.) |
Ref | Expression |
---|---|
uspgrloopvtx.g | ⊢ 𝐺 = 〈𝑉, {〈𝐴, {𝑁}〉}〉 |
Ref | Expression |
---|---|
uspgrloopedg | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = {{𝑁}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uspgrloopvtx.g | . . . 4 ⊢ 𝐺 = 〈𝑉, {〈𝐴, {𝑁}〉}〉 | |
2 | 1 | fveq2i 6648 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) |
3 | snex 5297 | . . . . 5 ⊢ {〈𝐴, {𝑁}〉} ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → {〈𝐴, {𝑁}〉} ∈ V) |
5 | edgopval 26844 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ {〈𝐴, {𝑁}〉} ∈ V) → (Edg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) = ran {〈𝐴, {𝑁}〉}) | |
6 | 4, 5 | sylan2 595 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) = ran {〈𝐴, {𝑁}〉}) |
7 | 2, 6 | syl5eq 2845 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = ran {〈𝐴, {𝑁}〉}) |
8 | rnsnopg 6045 | . . 3 ⊢ (𝐴 ∈ 𝑋 → ran {〈𝐴, {𝑁}〉} = {{𝑁}}) | |
9 | 8 | adantl 485 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → ran {〈𝐴, {𝑁}〉} = {{𝑁}}) |
10 | 7, 9 | eqtrd 2833 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = {{𝑁}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 〈cop 4531 ran crn 5520 ‘cfv 6324 Edgcedg 26840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-2nd 7672 df-iedg 26792 df-edg 26841 |
This theorem is referenced by: (None) |
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