| Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | ||
| 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 29197 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph) | |
| 2 | uspgrushgr 29194 | . . 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 29246 | . 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 1540 ∈ wcel 2108 {crab 3436 ↦ cmpt 5225 dom cdm 5685 –1-1-onto→wf1o 6560 ‘cfv 6561 Vtxcvtx 29013 iEdgciedg 29014 Edgcedg 29064 USHGraphcushgr 29074 USPGraphcuspgr 29165 USGraphcusgr 29166 | 
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-i2m1 11223 ax-1ne0 11224 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 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-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-2 12329 df-edg 29065 df-uhgr 29075 df-ushgr 29076 df-uspgr 29167 df-usgr 29168 | 
| This theorem is referenced by: usgredgleordALT 29251 | 
| Copyright terms: Public domain | W3C validator |