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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 28171 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐺 ∈ USPGraph) | |
2 | uspgrushgr 28168 | . . 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 28219 | . 2 ⊢ ((𝐺 ∈ USHGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵) |
11 | 3, 10 | sylan 581 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉) → 𝐹:𝐴–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3406 ↦ cmpt 5189 dom cdm 5634 –1-1-onto→wf1o 6496 ‘cfv 6497 Vtxcvtx 27989 iEdgciedg 27990 Edgcedg 28040 USHGraphcushgr 28050 USPGraphcuspgr 28141 USGraphcusgr 28142 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-i2m1 11124 ax-1ne0 11125 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-2 12221 df-edg 28041 df-uhgr 28051 df-ushgr 28052 df-uspgr 28143 df-usgr 28144 |
This theorem is referenced by: usgredgleordALT 28224 |
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