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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isubgriedg | Structured version Visualization version GIF version |
Description: The edges of an induced subgraph. (Contributed by AV, 12-May-2025.) |
Ref | Expression |
---|---|
isubgriedg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isubgriedg.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
isubgriedg | ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isubgriedg.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | isubgriedg.e | . . . 4 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | isisubgr 47786 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) = 〈𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})〉) |
4 | 3 | fveq2d 6911 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = (iEdg‘〈𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})〉)) |
5 | 1 | fvexi 6921 | . . . 4 ⊢ 𝑉 ∈ V |
6 | 5 | ssex 5327 | . . 3 ⊢ (𝑆 ⊆ 𝑉 → 𝑆 ∈ V) |
7 | 2 | fvexi 6921 | . . . . 5 ⊢ 𝐸 ∈ V |
8 | 7 | a1i 11 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝐸 ∈ V) |
9 | 8 | resexd 6048 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆}) ∈ V) |
10 | opiedgfv 29039 | . . 3 ⊢ ((𝑆 ∈ V ∧ (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆}) ∈ V) → (iEdg‘〈𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})〉) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})) | |
11 | 6, 9, 10 | syl2an2 686 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘〈𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})〉) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})) |
12 | 4, 11 | eqtrd 2775 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ⊆ wss 3963 〈cop 4637 dom cdm 5689 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 Vtxcvtx 29028 iEdgciedg 29029 ISubGr cisubgr 47784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8014 df-iedg 29031 df-isubgr 47785 |
This theorem is referenced by: isubgredgss 47789 isubgredg 47790 isubgruhgr 47792 isubgrsubgr 47793 isubgrgrim 47835 |
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