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| Description: Lemma 3 for upgrres1 29331. (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 6908 | . 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 29327 | . . 3 ⊢ ((𝑉 ∖ {𝑁}) ∈ V ∧ ( I ↾ 𝐹) ∈ V) | 
| 7 | opiedgfv 29025 | . . 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 2764 | 1 ⊢ (iEdg‘𝑆) = ( I ↾ 𝐹) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∉ wnel 3045 {crab 3435 Vcvv 3479 ∖ cdif 3947 {csn 4625 〈cop 4631 I cid 5576 ↾ cres 5686 ‘cfv 6560 Vtxcvtx 29014 iEdgciedg 29015 Edgcedg 29065 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-iota 6513 df-fun 6562 df-fv 6568 df-2nd 8016 df-iedg 29017 | 
| This theorem is referenced by: upgrres1 29331 umgrres1 29332 usgrres1 29333 nbupgrres 29382 | 
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