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Mirrors > Home > MPE Home > Th. List > usgredgedg | Structured version Visualization version GIF version |
Description: In a simple graph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.) |
Ref | Expression |
---|---|
ushgredgedg.e | ⊢ 𝐸 = (Edg‘𝐺) |
ushgredgedg.i | ⊢ 𝐼 = (iEdg‘𝐺) |
ushgredgedg.v | ⊢ 𝑉 = (Vtx‘𝐺) |
ushgredgedg.a | ⊢ 𝐴 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} |
ushgredgedg.b | ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} |
ushgredgedg.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥)) |
Ref | Expression |
---|---|
usgredgedg | ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgruspgr 29006 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph) | |
2 | uspgrushgr 29003 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ USHGraph) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USHGraph) |
4 | ushgredgedg.e | . . 3 ⊢ 𝐸 = (Edg‘𝐺) | |
5 | ushgredgedg.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
6 | ushgredgedg.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
7 | ushgredgedg.a | . . 3 ⊢ 𝐴 = {𝑖 ∈ dom 𝐼 ∣ 𝑁 ∈ (𝐼‘𝑖)} | |
8 | ushgredgedg.b | . . 3 ⊢ 𝐵 = {𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒} | |
9 | ushgredgedg.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐼‘𝑥)) | |
10 | 4, 5, 6, 7, 8, 9 | ushgredgedg 29055 | . 2 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵) |
11 | 3, 10 | sylan 579 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {crab 3429 ↦ cmpt 5231 dom cdm 5678 –1-1-onto→wf1o 6547 ‘cfv 6548 Vtxcvtx 28822 iEdgciedg 28823 Edgcedg 28873 USHGraphcushgr 28883 USPGraphcuspgr 28974 USGraphcusgr 28975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-i2m1 11207 ax-1ne0 11208 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-2 12306 df-edg 28874 df-uhgr 28884 df-ushgr 28885 df-uspgr 28976 df-usgr 28977 |
This theorem is referenced by: usgredgleordALT 29060 |
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