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| Mirrors > Home > MPE Home > Th. List > issubgr2 | Structured version Visualization version GIF version | ||
| Description: The property of a set to be a subgraph of a set whose edge function is actually a function. (Contributed by AV, 20-Nov-2020.) |
| Ref | Expression |
|---|---|
| issubgr.v | ⊢ 𝑉 = (Vtx‘𝑆) |
| issubgr.a | ⊢ 𝐴 = (Vtx‘𝐺) |
| issubgr.i | ⊢ 𝐼 = (iEdg‘𝑆) |
| issubgr.b | ⊢ 𝐵 = (iEdg‘𝐺) |
| issubgr.e | ⊢ 𝐸 = (Edg‘𝑆) |
| Ref | Expression |
|---|---|
| issubgr2 | ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | issubgr.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝑆) | |
| 2 | issubgr.a | . . . 4 ⊢ 𝐴 = (Vtx‘𝐺) | |
| 3 | issubgr.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝑆) | |
| 4 | issubgr.b | . . . 4 ⊢ 𝐵 = (iEdg‘𝐺) | |
| 5 | issubgr.e | . . . 4 ⊢ 𝐸 = (Edg‘𝑆) | |
| 6 | 1, 2, 3, 4, 5 | issubgr 29288 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))) |
| 7 | 6 | 3adant2 1132 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))) |
| 8 | resss 6019 | . . . . 5 ⊢ (𝐵 ↾ dom 𝐼) ⊆ 𝐵 | |
| 9 | sseq1 4009 | . . . . 5 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼 ⊆ 𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵)) | |
| 10 | 8, 9 | mpbiri 258 | . . . 4 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼 ⊆ 𝐵) |
| 11 | funssres 6610 | . . . . . . 7 ⊢ ((Fun 𝐵 ∧ 𝐼 ⊆ 𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼) | |
| 12 | 11 | eqcomd 2743 | . . . . . 6 ⊢ ((Fun 𝐵 ∧ 𝐼 ⊆ 𝐵) → 𝐼 = (𝐵 ↾ dom 𝐼)) |
| 13 | 12 | ex 412 | . . . . 5 ⊢ (Fun 𝐵 → (𝐼 ⊆ 𝐵 → 𝐼 = (𝐵 ↾ dom 𝐼))) |
| 14 | 13 | 3ad2ant2 1135 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝐼 ⊆ 𝐵 → 𝐼 = (𝐵 ↾ dom 𝐼))) |
| 15 | 10, 14 | impbid2 226 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝐼 = (𝐵 ↾ dom 𝐼) ↔ 𝐼 ⊆ 𝐵)) |
| 16 | 15 | 3anbi2d 1443 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → ((𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉) ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉))) |
| 17 | 7, 16 | bitrd 279 | 1 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 ⊆ 𝐵 ∧ 𝐸 ⊆ 𝒫 𝑉))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 𝒫 cpw 4600 class class class wbr 5143 dom cdm 5685 ↾ cres 5687 Fun wfun 6555 ‘cfv 6561 Vtxcvtx 29013 iEdgciedg 29014 Edgcedg 29064 SubGraph csubgr 29284 |
| 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-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 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-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-subgr 29285 |
| This theorem is referenced by: uhgrspansubgr 29308 |
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