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Mirrors > Home > MPE Home > Th. List > subgreldmiedg | Structured version Visualization version GIF version |
Description: An element of the domain of the edge function of a subgraph is an element of the domain of the edge function of the supergraph. (Contributed by AV, 20-Nov-2020.) |
Ref | Expression |
---|---|
subgreldmiedg | ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆) | |
2 | eqid 2734 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
3 | eqid 2734 | . . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆) | |
4 | eqid 2734 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
5 | eqid 2734 | . . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆) | |
6 | 1, 2, 3, 4, 5 | subgrprop2 27334 | . . 3 ⊢ (𝑆 SubGraph 𝐺 → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))) |
7 | dmss 5760 | . . . . 5 ⊢ ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → dom (iEdg‘𝑆) ⊆ dom (iEdg‘𝐺)) | |
8 | 7 | 3ad2ant2 1136 | . . . 4 ⊢ (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → dom (iEdg‘𝑆) ⊆ dom (iEdg‘𝐺)) |
9 | 8 | sseld 3890 | . . 3 ⊢ (((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) → (𝑋 ∈ dom (iEdg‘𝑆) → 𝑋 ∈ dom (iEdg‘𝐺))) |
10 | 6, 9 | syl 17 | . 2 ⊢ (𝑆 SubGraph 𝐺 → (𝑋 ∈ dom (iEdg‘𝑆) → 𝑋 ∈ dom (iEdg‘𝐺))) |
11 | 10 | imp 410 | 1 ⊢ ((𝑆 SubGraph 𝐺 ∧ 𝑋 ∈ dom (iEdg‘𝑆)) → 𝑋 ∈ dom (iEdg‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2110 ⊆ wss 3857 𝒫 cpw 4503 class class class wbr 5043 dom cdm 5540 ‘cfv 6369 Vtxcvtx 27059 iEdgciedg 27060 Edgcedg 27110 SubGraph csubgr 27327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-xp 5546 df-rel 5547 df-dm 5550 df-res 5552 df-iota 6327 df-fv 6377 df-subgr 27328 |
This theorem is referenced by: subgruhgredgd 27344 subumgredg2 27345 subupgr 27347 |
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