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Mirrors > Home > MPE Home > Th. List > vtxvalsnop | Structured version Visualization version GIF version |
Description: Degenerated case 2 for vertices: The set of vertices of a singleton containing an ordered pair with equal components is the singleton containing the component. (Contributed by AV, 24-Sep-2020.) (Proof shortened by AV, 15-Jul-2022.) (Avoid depending on this detail.) |
Ref | Expression |
---|---|
vtxvalsnop.b | ⊢ 𝐵 ∈ V |
vtxvalsnop.g | ⊢ 𝐺 = {〈𝐵, 𝐵〉} |
Ref | Expression |
---|---|
vtxvalsnop | ⊢ (Vtx‘𝐺) = {𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtxvalsnop.g | . . 3 ⊢ 𝐺 = {〈𝐵, 𝐵〉} | |
2 | 1 | fveq2i 6899 | . 2 ⊢ (Vtx‘𝐺) = (Vtx‘{〈𝐵, 𝐵〉}) |
3 | vtxvalsnop.b | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 3 | snopeqopsnid 5511 | . . 3 ⊢ {〈𝐵, 𝐵〉} = 〈{𝐵}, {𝐵}〉 |
5 | 4 | fveq2i 6899 | . 2 ⊢ (Vtx‘{〈𝐵, 𝐵〉}) = (Vtx‘〈{𝐵}, {𝐵}〉) |
6 | snex 5433 | . . 3 ⊢ {𝐵} ∈ V | |
7 | 6, 6 | opvtxfvi 28894 | . 2 ⊢ (Vtx‘〈{𝐵}, {𝐵}〉) = {𝐵} |
8 | 2, 5, 7 | 3eqtri 2757 | 1 ⊢ (Vtx‘𝐺) = {𝐵} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3461 {csn 4630 〈cop 4636 ‘cfv 6549 Vtxcvtx 28881 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6501 df-fun 6551 df-fv 6557 df-1st 7994 df-vtx 28883 |
This theorem is referenced by: vtxval3sn 28928 |
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