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Mirrors > Home > MPE Home > Th. List > upgrres1lem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for upgrres1 27097. (Contributed by AV, 7-Nov-2020.) |
Ref | Expression |
---|---|
upgrres1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
upgrres1.e | ⊢ 𝐸 = (Edg‘𝐺) |
upgrres1.f | ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} |
upgrres1.s | ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 |
Ref | Expression |
---|---|
upgrres1lem2 | ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgrres1.s | . . 3 ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 | |
2 | 1 | fveq2i 6675 | . 2 ⊢ (Vtx‘𝑆) = (Vtx‘〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉) |
3 | upgrres1.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | upgrres1.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
5 | upgrres1.f | . . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} | |
6 | 3, 4, 5 | upgrres1lem1 27093 | . . 3 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) |
7 | opvtxfv 26791 | . . 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 2846 | 1 ⊢ (Vtx‘𝑆) = (𝑉 ∖ {𝑁}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∉ wnel 3125 {crab 3144 Vcvv 3496 ∖ cdif 3935 {csn 4569 〈cop 4575 I cid 5461 ↾ cres 5559 ‘cfv 6357 Vtxcvtx 26783 Edgcedg 26834 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-iota 6316 df-fun 6359 df-fv 6365 df-1st 7691 df-vtx 26785 |
This theorem is referenced by: upgrres1 27097 umgrres1 27098 usgrres1 27099 nbupgrres 27148 nbupgruvtxres 27191 uvtxupgrres 27192 cusgrres 27232 |
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