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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 26958) 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 6666 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) |
3 | snex 5322 | . . . . 5 ⊢ {〈𝐴, {𝑁}〉} ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → {〈𝐴, {𝑁}〉} ∈ V) |
5 | edgopval 26763 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ {〈𝐴, {𝑁}〉} ∈ V) → (Edg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) = ran {〈𝐴, {𝑁}〉}) | |
6 | 4, 5 | sylan2 592 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) = ran {〈𝐴, {𝑁}〉}) |
7 | 2, 6 | syl5eq 2865 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = ran {〈𝐴, {𝑁}〉}) |
8 | rnsnopg 6071 | . . 3 ⊢ (𝐴 ∈ 𝑋 → ran {〈𝐴, {𝑁}〉} = {{𝑁}}) | |
9 | 8 | adantl 482 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → ran {〈𝐴, {𝑁}〉} = {{𝑁}}) |
10 | 7, 9 | eqtrd 2853 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = {{𝑁}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 〈cop 4563 ran crn 5549 ‘cfv 6348 Edgcedg 26759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-2nd 7679 df-iedg 26711 df-edg 26760 |
This theorem is referenced by: (None) |
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