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Mirrors > Home > MPE Home > Th. List > uhgrspan1lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for uhgrspan1 28300. (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 6860 | . . 3 ⊢ 𝑉 ∈ V |
3 | 2 | difexi 5289 | . 2 ⊢ (𝑉 ∖ {𝑁}) ∈ V |
4 | uhgrspan1.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
5 | 4 | fvexi 6860 | . . 3 ⊢ 𝐼 ∈ V |
6 | 5 | resex 5989 | . 2 ⊢ (𝐼 ↾ 𝐹) ∈ V |
7 | 3, 6 | pm3.2i 472 | 1 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼 ↾ 𝐹) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∉ wnel 3046 {crab 3406 Vcvv 3447 ∖ cdif 3911 {csn 4590 dom cdm 5637 ↾ cres 5639 ‘cfv 6500 Vtxcvtx 27996 iEdgciedg 27997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-sn 4591 df-pr 4593 df-uni 4870 df-res 5649 df-iota 6452 df-fv 6508 |
This theorem is referenced by: uhgrspan1lem2 28298 uhgrspan1lem3 28299 |
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