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Mirrors > Home > MPE Home > Th. List > uhgrspan1lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for uhgrspan1 26195. (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 6194 | . 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 26192 | . . 3 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ (𝐼 ↾ 𝐹) ∈ V) |
7 | opiedgfv 25887 | . . 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 2644 | 1 ⊢ (iEdg‘𝑆) = (𝐼 ↾ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∉ wnel 2897 {crab 2916 Vcvv 3200 ∖ cdif 3571 {csn 4177 〈cop 4183 dom cdm 5114 ↾ cres 5116 ‘cfv 5888 Vtxcvtx 25874 iEdgciedg 25875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 df-2nd 7169 df-iedg 25877 |
This theorem is referenced by: uhgrspan1 26195 upgrres 26198 umgrres 26199 usgrres 26200 |
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