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Mirrors > Home > MPE Home > Th. List > uhgrspan1lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for uhgrspan1 27012. (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 6677 | . . 3 ⊢ 𝑉 ∈ V |
3 | 2 | difexi 5223 | . 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V |
4 | uhgrspan1.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
5 | 4 | fvexi 6677 | . . 3 ⊢ 𝐼 ∈ V |
6 | 5 | resex 5892 | . 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 1528 ∈ wcel 2105 ∉ wnel 3120 {crab 3139 Vcvv 3492 ∖ cdif 3930 {csn 4557 dom cdm 5548 ↾ cres 5550 ‘cfv 6348 Vtxcvtx 26708 iEdgciedg 26709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-sn 4558 df-pr 4560 df-uni 4831 df-res 5560 df-iota 6307 df-fv 6356 |
This theorem is referenced by: uhgrspan1lem2 27010 uhgrspan1lem3 27011 |
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