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Mirrors > Home > MPE Home > Th. List > uhgrspan1lem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for uhgrspan1 27071. (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 6659 | . 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 27068 | . . 3 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼 ↾ 𝐹) ∈ V) |
7 | opvtxfv 26775 | . . 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 2844 | 1 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∉ wnel 3123 {crab 3142 Vcvv 3486 ∖ cdif 3921 {csn 4553 〈cop 4559 dom cdm 5541 ↾ cres 5543 ‘cfv 6341 Vtxcvtx 26767 iEdgciedg 26768 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-iota 6300 df-fun 6343 df-fv 6349 df-1st 7675 df-vtx 26769 |
This theorem is referenced by: uhgrspan1 27071 upgrres 27074 umgrres 27075 usgrres 27076 |
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