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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 26547) 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 6437 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) |
3 | snex 5130 | . . . . 5 ⊢ {〈𝐴, {𝑁}〉} ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → {〈𝐴, {𝑁}〉} ∈ V) |
5 | edgopval 26350 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ {〈𝐴, {𝑁}〉} ∈ V) → (Edg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) = ran {〈𝐴, {𝑁}〉}) | |
6 | 4, 5 | sylan2 588 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) = ran {〈𝐴, {𝑁}〉}) |
7 | 2, 6 | syl5eq 2874 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = ran {〈𝐴, {𝑁}〉}) |
8 | rnsnopg 5856 | . . 3 ⊢ (𝐴 ∈ 𝑋 → ran {〈𝐴, {𝑁}〉} = {{𝑁}}) | |
9 | 8 | adantl 475 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → ran {〈𝐴, {𝑁}〉} = {{𝑁}}) |
10 | 7, 9 | eqtrd 2862 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = {{𝑁}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 Vcvv 3415 {csn 4398 〈cop 4404 ran crn 5344 ‘cfv 6124 Edgcedg 26346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-iota 6087 df-fun 6126 df-fv 6132 df-2nd 7430 df-iedg 26298 df-edg 26347 |
This theorem is referenced by: (None) |
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