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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 29182) 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 6896 | . . 3 ⊢ (Edg‘𝐺) = (Edg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) |
3 | snex 5429 | . . . . 5 ⊢ {〈𝐴, {𝑁}〉} ∈ V | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → {〈𝐴, {𝑁}〉} ∈ V) |
5 | edgopval 28984 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ {〈𝐴, {𝑁}〉} ∈ V) → (Edg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) = ran {〈𝐴, {𝑁}〉}) | |
6 | 4, 5 | sylan2 591 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘〈𝑉, {〈𝐴, {𝑁}〉}〉) = ran {〈𝐴, {𝑁}〉}) |
7 | 2, 6 | eqtrid 2778 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = ran {〈𝐴, {𝑁}〉}) |
8 | rnsnopg 6224 | . . 3 ⊢ (𝐴 ∈ 𝑋 → ran {〈𝐴, {𝑁}〉} = {{𝑁}}) | |
9 | 8 | adantl 480 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → ran {〈𝐴, {𝑁}〉} = {{𝑁}}) |
10 | 7, 9 | eqtrd 2766 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) → (Edg‘𝐺) = {{𝑁}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 {csn 4623 〈cop 4629 ran crn 5675 ‘cfv 6546 Edgcedg 28980 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-iota 6498 df-fun 6548 df-fv 6554 df-2nd 7996 df-iedg 28932 df-edg 28981 |
This theorem is referenced by: (None) |
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