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Mirrors > Home > MPE Home > Th. List > Mathboxes > uhgrspan1lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for uhgrspan1 40527. (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 |
---|---|
uhgrspan1lem3 | ⊢ (iEdg‘𝑆) = (𝐼 ↾ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrspan1.s | . . 3 ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)〉 | |
2 | 1 | fveq2i 6106 | . 2 ⊢ (iEdg‘𝑆) = (iEdg‘〈(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 40524 | . . 3 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼 ↾ 𝐹) ∈ V) |
7 | opiedgfv 25684 | . . 3 ⊢ (((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼 ↾ 𝐹) ∈ V) → (iEdg‘〈(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)〉) = (𝐼 ↾ 𝐹)) | |
8 | 6, 7 | ax-mp 5 | . 2 ⊢ (iEdg‘〈(𝑉 ∖ {𝑁}), (𝐼 ↾ 𝐹)〉) = (𝐼 ↾ 𝐹) |
9 | 2, 8 | eqtri 2632 | 1 ⊢ (iEdg‘𝑆) = (𝐼 ↾ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∉ wnel 2781 {crab 2900 Vcvv 3173 ∖ cdif 3537 {csn 4125 〈cop 4131 dom cdm 5038 ↾ cres 5040 ‘cfv 5804 Vtxcvtx 25673 iEdgciedg 25674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fun 5806 df-fv 5812 df-2nd 7060 df-iedg 25676 |
This theorem is referenced by: uhgrspan1 40527 |
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