| 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 2737 | . . . . . 6 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
| 5 | 3, 4 | isubgriedg 47849 | . . . . 5 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})) |
| 6 | 2, 5 | eqtrid 2789 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (iEdg‘𝐻) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})) |
| 7 | 6 | rneqd 5949 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → ran (iEdg‘𝐻) = ran ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})) |
| 8 | resss 6019 | . . . 4 ⊢ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (iEdg‘𝐺) | |
| 9 | rnss 5950 | . . . 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 4018 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → ran (iEdg‘𝐻) ⊆ ran (iEdg‘𝐺)) |
| 12 | isubgredg.i | . . 3 ⊢ 𝐼 = (Edg‘𝐻) | |
| 13 | edgval 29066 | . . 3 ⊢ (Edg‘𝐻) = ran (iEdg‘𝐻) | |
| 14 | 12, 13 | eqtri 2765 | . 2 ⊢ 𝐼 = ran (iEdg‘𝐻) |
| 15 | isubgredg.e | . . 3 ⊢ 𝐸 = (Edg‘𝐺) | |
| 16 | edgval 29066 | . . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
| 17 | 15, 16 | eqtri 2765 | . 2 ⊢ 𝐸 = ran (iEdg‘𝐺) |
| 18 | 11, 14, 17 | 3sstr4g 4037 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝐼 ⊆ 𝐸) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 ⊆ wss 3951 dom cdm 5685 ran crn 5686 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 Vtxcvtx 29013 iEdgciedg 29014 Edgcedg 29064 ISubGr cisubgr 47846 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8015 df-iedg 29016 df-edg 29065 df-isubgr 47847 |
| This theorem is referenced by: (None) |
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