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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isubgredgss | Structured version Visualization version GIF version |
Description: The edges of an induced subgraph of a graph are edges of the graph. (Contributed by AV, 24-Sep-2025.) |
Ref | Expression |
---|---|
isubgredg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isubgredg.e | ⊢ 𝐸 = (Edg‘𝐺) |
isubgredg.h | ⊢ 𝐻 = (𝐺 ISubGr 𝑆) |
isubgredg.i | ⊢ 𝐼 = (Edg‘𝐻) |
Ref | Expression |
---|---|
isubgredgss | ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝐼 ⊆ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isubgredg.h | . . . . . 6 ⊢ 𝐻 = (𝐺 ISubGr 𝑆) | |
2 | 1 | fveq2i 6909 | . . . . 5 ⊢ (iEdg‘𝐻) = (iEdg‘(𝐺 ISubGr 𝑆)) |
3 | isubgredg.v | . . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺) | |
4 | eqid 2734 | . . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
5 | 3, 4 | isubgriedg 47786 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})) |
6 | 2, 5 | eqtrid 2786 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘𝐻) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})) |
7 | 6 | rneqd 5951 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → ran (iEdg‘𝐻) = ran ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})) |
8 | resss 6021 | . . . 4 ⊢ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (iEdg‘𝐺) | |
9 | rnss 5952 | . . . 4 ⊢ (((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (iEdg‘𝐺) → ran ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ ran (iEdg‘𝐺)) | |
10 | 8, 9 | mp1i 13 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → ran ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ ran (iEdg‘𝐺)) |
11 | 7, 10 | eqsstrd 4033 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → ran (iEdg‘𝐻) ⊆ ran (iEdg‘𝐺)) |
12 | isubgredg.i | . . 3 ⊢ 𝐼 = (Edg‘𝐻) | |
13 | edgval 29080 | . . 3 ⊢ (Edg‘𝐻) = ran (iEdg‘𝐻) | |
14 | 12, 13 | eqtri 2762 | . 2 ⊢ 𝐼 = ran (iEdg‘𝐻) |
15 | isubgredg.e | . . 3 ⊢ 𝐸 = (Edg‘𝐺) | |
16 | edgval 29080 | . . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
17 | 15, 16 | eqtri 2762 | . 2 ⊢ 𝐸 = ran (iEdg‘𝐺) |
18 | 11, 14, 17 | 3sstr4g 4040 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝐼 ⊆ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 {crab 3432 ⊆ wss 3962 dom cdm 5688 ran crn 5689 ↾ cres 5690 ‘cfv 6562 (class class class)co 7430 Vtxcvtx 29027 iEdgciedg 29028 Edgcedg 29078 ISubGr cisubgr 47783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-2nd 8013 df-iedg 29030 df-edg 29079 df-isubgr 47784 |
This theorem is referenced by: (None) |
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