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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 26575 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))) |
7 | 6 | 3adant2 1165 | . 2 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ (𝑉 ⊆ 𝐴 ∧ 𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))) |
8 | resss 5662 | . . . . 5 ⊢ (𝐵 ↾ dom 𝐼) ⊆ 𝐵 | |
9 | sseq1 3851 | . . . . 5 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼 ⊆ 𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵)) | |
10 | 8, 9 | mpbiri 250 | . . . 4 ⊢ (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼 ⊆ 𝐵) |
11 | funssres 6170 | . . . . . . 7 ⊢ ((Fun 𝐵 ∧ 𝐼 ⊆ 𝐵) → (𝐵 ↾ dom 𝐼) = 𝐼) | |
12 | 11 | eqcomd 2831 | . . . . . 6 ⊢ ((Fun 𝐵 ∧ 𝐼 ⊆ 𝐵) → 𝐼 = (𝐵 ↾ dom 𝐼)) |
13 | 12 | ex 403 | . . . . 5 ⊢ (Fun 𝐵 → (𝐼 ⊆ 𝐵 → 𝐼 = (𝐵 ↾ dom 𝐼))) |
14 | 13 | 3ad2ant2 1168 | . . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝐼 ⊆ 𝐵 → 𝐼 = (𝐵 ↾ dom 𝐼))) |
15 | 10, 14 | impbid2 218 | . . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ Fun 𝐵 ∧ 𝑆 ∈ 𝑈) → (𝐼 = (𝐵 ↾ dom 𝐼) ↔ 𝐼 ⊆ 𝐵)) |
16 | 15 | 3anbi2d 1569 | . 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 271 | 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 198 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 𝒫 cpw 4380 class class class wbr 4875 dom cdm 5346 ↾ cres 5348 Fun wfun 6121 ‘cfv 6127 Vtxcvtx 26301 iEdgciedg 26302 Edgcedg 26352 SubGraph csubgr 26571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-res 5358 df-iota 6090 df-fun 6129 df-fv 6135 df-subgr 26572 |
This theorem is referenced by: uhgrspansubgr 26595 |
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