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Mirrors > Home > MPE Home > Th. List > upgrres1lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for upgrres1 27373. (Contributed by AV, 7-Nov-2020.) |
Ref | Expression |
---|---|
upgrres1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
upgrres1.e | ⊢ 𝐸 = (Edg‘𝐺) |
upgrres1.f | ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} |
upgrres1.s | ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 |
Ref | Expression |
---|---|
upgrres1lem3 | ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgrres1.s | . . 3 ⊢ 𝑆 = 〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉 | |
2 | 1 | fveq2i 6709 | . 2 ⊢ (iEdg‘𝑆) = (iEdg‘〈(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)〉) |
3 | upgrres1.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | upgrres1.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
5 | upgrres1.f | . . . 4 ⊢ 𝐹 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∉ 𝑒} | |
6 | 3, 4, 5 | upgrres1lem1 27369 | . . 3 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) |
7 | opiedgfv 27070 | . . 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 2762 | 1 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∉ wnel 3039 {crab 3058 Vcvv 3401 ∖ cdif 3854 {csn 4531 〈cop 4537 I cid 5443 ↾ cres 5542 ‘cfv 6369 Vtxcvtx 27059 iEdgciedg 27060 Edgcedg 27110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-iota 6327 df-fun 6371 df-fv 6377 df-2nd 7751 df-iedg 27062 |
This theorem is referenced by: upgrres1 27373 umgrres1 27374 usgrres1 27375 nbupgrres 27424 |
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