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Mirrors > Home > MPE Home > Th. List > vtxdginducedm1lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for vtxdginducedm1 28491: the edge function 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 |
---|---|
vtxdginducedm1lem1 | ⊢ (iEdg‘𝑆) = 𝑃 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vtxdginducedm1.s | . . 3 ⊢ 𝑆 = 〈𝐾, 𝑃〉 | |
2 | 1 | fveq2i 6845 | . 2 ⊢ (iEdg‘𝑆) = (iEdg‘〈𝐾, 𝑃〉) |
3 | vtxdginducedm1.k | . . . 4 ⊢ 𝐾 = (𝑉 ∖ {𝑁}) | |
4 | vtxdginducedm1.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
5 | 4 | fvexi 6856 | . . . . 5 ⊢ 𝑉 ∈ V |
6 | 5 | difexi 5285 | . . . 4 ⊢ (𝑉 ∖ {𝑁}) ∈ V |
7 | 3, 6 | eqeltri 2834 | . . 3 ⊢ 𝐾 ∈ V |
8 | vtxdginducedm1.p | . . . 4 ⊢ 𝑃 = (𝐸 ↾ 𝐼) | |
9 | vtxdginducedm1.e | . . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺) | |
10 | 9 | fvexi 6856 | . . . . 5 ⊢ 𝐸 ∈ V |
11 | 10 | resex 5985 | . . . 4 ⊢ (𝐸 ↾ 𝐼) ∈ V |
12 | 8, 11 | eqeltri 2834 | . . 3 ⊢ 𝑃 ∈ V |
13 | 7, 12 | opiedgfvi 27961 | . 2 ⊢ (iEdg‘〈𝐾, 𝑃〉) = 𝑃 |
14 | 2, 13 | eqtri 2764 | 1 ⊢ (iEdg‘𝑆) = 𝑃 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∉ wnel 3049 {crab 3407 Vcvv 3445 ∖ cdif 3907 {csn 4586 〈cop 4592 dom cdm 5633 ↾ cres 5635 ‘cfv 6496 Vtxcvtx 27947 iEdgciedg 27948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-iota 6448 df-fun 6498 df-fv 6504 df-2nd 7922 df-iedg 27950 |
This theorem is referenced by: vtxdginducedm1lem2 28488 vtxdginducedm1lem3 28489 vtxdginducedm1fi 28492 finsumvtxdg2ssteplem4 28496 |
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