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Mirrors > Home > MPE Home > Th. List > uhgrspan1lem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for uhgrspan1 27093. (Contributed by AV, 19-Nov-2020.) |
Ref | Expression |
---|---|
uhgrspan1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
uhgrspan1.i | ⊢ 𝐼 = (iEdg‘𝐺) |
uhgrspan1.f | ⊢ 𝐹 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∉ (𝐼‘𝑖)} |
uhgrspan1.s | ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)〉 |
Ref | Expression |
---|---|
uhgrspan1lem2 | ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrspan1.s | . . 3 ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)〉 | |
2 | 1 | fveq2i 6648 | . 2 ⊢ (Vtx‘𝑆) = (Vtx‘〈(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)〉) |
3 | uhgrspan1.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | uhgrspan1.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
5 | uhgrspan1.f | . . . 4 ⊢ 𝐹 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∉ (𝐼‘𝑖)} | |
6 | 3, 4, 5 | uhgrspan1lem1 27090 | . . 3 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼 ↾ 𝐹) ∈ V) |
7 | opvtxfv 26797 | . . 3 ⊢ (((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼 ↾ 𝐹) ∈ V) → (Vtx‘〈(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)〉) = (𝑉 ∖ {𝑁})) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ (Vtx‘〈(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)〉) = (𝑉 ∖ {𝑁}) |
9 | 2, 8 | eqtri 2821 | 1 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∉ wnel 3091 {crab 3110 Vcvv 3441 ∖ cdif 3878 {csn 4525 〈cop 4531 dom cdm 5519 ↾ cres 5521 ‘cfv 6324 Vtxcvtx 26789 iEdgciedg 26790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-fv 6332 df-1st 7671 df-vtx 26791 |
This theorem is referenced by: uhgrspan1 27093 upgrres 27096 umgrres 27097 usgrres 27098 |
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