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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 6440 | . 2 ⊢ (Vtx‘𝐺) = (Vtx‘{〈𝐵, 𝐵〉}) |
3 | vtxvalsnop.b | . . . 4 ⊢ 𝐵 ∈ V | |
4 | 3 | snopeqopsnid 5197 | . . 3 ⊢ {〈𝐵, 𝐵〉} = 〈{𝐵}, {𝐵}〉 |
5 | 4 | fveq2i 6440 | . 2 ⊢ (Vtx‘{〈𝐵, 𝐵〉}) = (Vtx‘〈{𝐵}, {𝐵}〉) |
6 | snex 5131 | . . 3 ⊢ {𝐵} ∈ V | |
7 | 6, 6 | opvtxfvi 26314 | . 2 ⊢ (Vtx‘〈{𝐵}, {𝐵}〉) = {𝐵} |
8 | 2, 5, 7 | 3eqtri 2853 | 1 ⊢ (Vtx‘𝐺) = {𝐵} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 Vcvv 3414 {csn 4399 〈cop 4405 ‘cfv 6127 Vtxcvtx 26301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-iota 6090 df-fun 6129 df-fv 6135 df-1st 7433 df-vtx 26303 |
This theorem is referenced by: vtxval3sn 26348 |
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