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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 29212 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph) | |
2 | uspgrushgr 29209 | . . 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 29261 | . 2 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵) |
11 | 3, 10 | sylan 580 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 ↦ cmpt 5231 dom cdm 5689 –1-1-onto→wf1o 6562 ‘cfv 6563 Vtxcvtx 29028 iEdgciedg 29029 Edgcedg 29079 USHGraphcushgr 29089 USPGraphcuspgr 29180 USGraphcusgr 29181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-i2m1 11221 ax-1ne0 11222 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-2 12327 df-edg 29080 df-uhgr 29090 df-ushgr 29091 df-uspgr 29182 df-usgr 29183 |
This theorem is referenced by: usgredgleordALT 29266 |
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