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Mirrors > Home > MPE Home > Th. List > uhgrspan1lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for uhgrspan1 28557. (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)} |
Ref | Expression |
---|---|
uhgrspan1lem1 | ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼 ↾ 𝐹) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrspan1.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | 1 | fvexi 6905 | . . 3 ⊢ 𝑉 ∈ V |
3 | 2 | difexi 5328 | . 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V |
4 | uhgrspan1.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
5 | 4 | fvexi 6905 | . . 3 ⊢ 𝐼 ∈ V |
6 | 5 | resex 6029 | . 2 ⊢ (𝐼 ↾ 𝐹) ∈ V |
7 | 3, 6 | pm3.2i 471 | 1 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼 ↾ 𝐹) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∉ wnel 3046 {crab 3432 Vcvv 3474 ∖ cdif 3945 {csn 4628 dom cdm 5676 ↾ cres 5678 ‘cfv 6543 Vtxcvtx 28253 iEdgciedg 28254 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-sn 4629 df-pr 4631 df-uni 4909 df-res 5688 df-iota 6495 df-fv 6551 |
This theorem is referenced by: uhgrspan1lem2 28555 uhgrspan1lem3 28556 |
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