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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 26965 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph) | |
2 | uspgrushgr 26962 | . . 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 27013 | . 2 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵) |
11 | 3, 10 | sylan 582 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3144 ↦ cmpt 5148 dom cdm 5557 –1-1-onto→wf1o 6356 ‘cfv 6357 Vtxcvtx 26783 iEdgciedg 26784 Edgcedg 26834 USHGraphcushgr 26844 USPGraphcuspgr 26935 USGraphcusgr 26936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-i2m1 10607 ax-1ne0 10608 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-2 11703 df-edg 26835 df-uhgr 26845 df-ushgr 26846 df-uspgr 26937 df-usgr 26938 |
This theorem is referenced by: usgredgleordALT 27018 |
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