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Mirrors > Home > MPE Home > Th. List > uspgrloopvtx | Structured version Visualization version GIF version |
Description: The set of vertices in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 28788). (Contributed by AV, 17-Dec-2020.) |
Ref | Expression |
---|---|
uspgrloopvtx.g | ⊢ 𝐺 = 〈𝑉, {〈𝐴, {𝑁}〉}〉 |
Ref | Expression |
---|---|
uspgrloopvtx | ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uspgrloopvtx.g | . . 3 ⊢ 𝐺 = 〈𝑉, {〈𝐴, {𝑁}〉}〉 | |
2 | 1 | fveq2i 6894 | . 2 ⊢ (Vtx‘𝐺) = (Vtx‘〈𝑉, {〈𝐴, {𝑁}〉}〉) |
3 | snex 5431 | . . 3 ⊢ {〈𝐴, {𝑁}〉} ∈ V | |
4 | opvtxfv 28546 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ {〈𝐴, {𝑁}〉} ∈ V) → (Vtx‘〈𝑉, {〈𝐴, {𝑁}〉}〉) = 𝑉) | |
5 | 3, 4 | mpan2 688 | . 2 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘〈𝑉, {〈𝐴, {𝑁}〉}〉) = 𝑉) |
6 | 2, 5 | eqtrid 2783 | 1 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 Vcvv 3473 {csn 4628 〈cop 4634 ‘cfv 6543 Vtxcvtx 28538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fv 6551 df-1st 7979 df-vtx 28540 |
This theorem is referenced by: uspgrloopvtxel 29055 uspgrloopnb0 29058 uspgrloopvd2 29059 |
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