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Mirrors > Home > MPE Home > Th. List > vtxdginducedm1lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for vtxdginducedm1 28199: an edge in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.) |
Ref | Expression |
---|---|
vtxdginducedm1.v | ⊢ 𝑉 = (Vtx‘𝐺) |
vtxdginducedm1.e | ⊢ 𝐸 = (iEdg‘𝐺) |
vtxdginducedm1.k | ⊢ 𝐾 = (𝑉 ∖ {𝑁}) |
vtxdginducedm1.i | ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} |
vtxdginducedm1.p | ⊢ 𝑃 = (𝐸 ↾ 𝐼) |
vtxdginducedm1.s | ⊢ 𝑆 = ⟨𝐾, 𝑃⟩ |
Ref | Expression |
---|---|
vtxdginducedm1lem3 | ⊢ (𝐻 ∈ 𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtxdginducedm1.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | vtxdginducedm1.e | . . . . 5 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | vtxdginducedm1.k | . . . . 5 ⊢ 𝐾 = (𝑉 ∖ {𝑁}) | |
4 | vtxdginducedm1.i | . . . . 5 ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)} | |
5 | vtxdginducedm1.p | . . . . 5 ⊢ 𝑃 = (𝐸 ↾ 𝐼) | |
6 | vtxdginducedm1.s | . . . . 5 ⊢ 𝑆 = ⟨𝐾, 𝑃⟩ | |
7 | 1, 2, 3, 4, 5, 6 | vtxdginducedm1lem1 28195 | . . . 4 ⊢ (iEdg‘𝑆) = 𝑃 |
8 | 7, 5 | eqtri 2764 | . . 3 ⊢ (iEdg‘𝑆) = (𝐸 ↾ 𝐼) |
9 | 8 | fveq1i 6826 | . 2 ⊢ ((iEdg‘𝑆)‘𝐻) = ((𝐸 ↾ 𝐼)‘𝐻) |
10 | fvres 6844 | . 2 ⊢ (𝐻 ∈ 𝐼 → ((𝐸 ↾ 𝐼)‘𝐻) = (𝐸‘𝐻)) | |
11 | 9, 10 | eqtrid 2788 | 1 ⊢ (𝐻 ∈ 𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ∉ wnel 3046 {crab 3403 ∖ cdif 3895 {csn 4573 ⟨cop 4579 dom cdm 5620 ↾ cres 5622 ‘cfv 6479 Vtxcvtx 27655 iEdgciedg 27656 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-iota 6431 df-fun 6481 df-fv 6487 df-2nd 7900 df-iedg 27658 |
This theorem is referenced by: vtxdginducedm1 28199 |
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