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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 27034). (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 6676 | . 2 ⊢ (Vtx‘𝐺) = (Vtx‘〈𝑉, {〈𝐴, {𝑁}〉}〉) |
3 | snex 5335 | . . 3 ⊢ {〈𝐴, {𝑁}〉} ∈ V | |
4 | opvtxfv 26792 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ {〈𝐴, {𝑁}〉} ∈ V) → (Vtx‘〈𝑉, {〈𝐴, {𝑁}〉}〉) = 𝑉) | |
5 | 3, 4 | mpan2 689 | . 2 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘〈𝑉, {〈𝐴, {𝑁}〉}〉) = 𝑉) |
6 | 2, 5 | syl5eq 2871 | 1 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3497 {csn 4570 〈cop 4576 ‘cfv 6358 Vtxcvtx 26784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fv 6366 df-1st 7692 df-vtx 26786 |
This theorem is referenced by: uspgrloopvtxel 27301 uspgrloopnb0 27304 uspgrloopvd2 27305 |
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