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Mirrors > Home > MPE Home > Th. List > snstriedgval | Structured version Visualization version GIF version |
Description: The set of indexed edges of a graph without edges represented as an extensible structure with vertices as base set and no indexed edges. See iedgvalsnop 26104 for the (degenerated) case where 𝑉 = (Base‘ndx). (Contributed by AV, 24-Sep-2020.) |
Ref | Expression |
---|---|
snstrvtxval.v | ⊢ 𝑉 ∈ V |
snstrvtxval.g | ⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉} |
Ref | Expression |
---|---|
snstriedgval | ⊢ (𝑉 ≠ (Base‘ndx) → (iEdg‘𝐺) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iedgval 26049 | . . 3 ⊢ (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑉 ≠ (Base‘ndx) → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺))) |
3 | necom 2973 | . . . 4 ⊢ (𝑉 ≠ (Base‘ndx) ↔ (Base‘ndx) ≠ 𝑉) | |
4 | fvex 6350 | . . . . 5 ⊢ (Base‘ndx) ∈ V | |
5 | snstrvtxval.v | . . . . 5 ⊢ 𝑉 ∈ V | |
6 | snstrvtxval.g | . . . . 5 ⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉} | |
7 | 4, 5, 6 | funsndifnop 6567 | . . . 4 ⊢ ((Base‘ndx) ≠ 𝑉 → ¬ 𝐺 ∈ (V × V)) |
8 | 3, 7 | sylbi 207 | . . 3 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ 𝐺 ∈ (V × V)) |
9 | 8 | iffalsed 4229 | . 2 ⊢ (𝑉 ≠ (Base‘ndx) → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (.ef‘𝐺)) |
10 | snex 5045 | . . . . . 6 ⊢ {〈(Base‘ndx), 𝑉〉} ∈ V | |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝐺 = {〈(Base‘ndx), 𝑉〉} → {〈(Base‘ndx), 𝑉〉} ∈ V) |
12 | 6, 11 | syl5eqel 2831 | . . . 4 ⊢ (𝐺 = {〈(Base‘ndx), 𝑉〉} → 𝐺 ∈ V) |
13 | edgfndxid 26041 | . . . 4 ⊢ (𝐺 ∈ V → (.ef‘𝐺) = (𝐺‘(.ef‘ndx))) | |
14 | 6, 12, 13 | mp2b 10 | . . 3 ⊢ (.ef‘𝐺) = (𝐺‘(.ef‘ndx)) |
15 | slotsbaseefdif 26043 | . . . . . . . 8 ⊢ (Base‘ndx) ≠ (.ef‘ndx) | |
16 | 15 | nesymi 2977 | . . . . . . 7 ⊢ ¬ (.ef‘ndx) = (Base‘ndx) |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ (.ef‘ndx) = (Base‘ndx)) |
18 | fvex 6350 | . . . . . . 7 ⊢ (.ef‘ndx) ∈ V | |
19 | 18 | elsn 4324 | . . . . . 6 ⊢ ((.ef‘ndx) ∈ {(Base‘ndx)} ↔ (.ef‘ndx) = (Base‘ndx)) |
20 | 17, 19 | sylnibr 318 | . . . . 5 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ (.ef‘ndx) ∈ {(Base‘ndx)}) |
21 | 6 | dmeqi 5468 | . . . . . 6 ⊢ dom 𝐺 = dom {〈(Base‘ndx), 𝑉〉} |
22 | dmsnopg 5753 | . . . . . . 7 ⊢ (𝑉 ∈ V → dom {〈(Base‘ndx), 𝑉〉} = {(Base‘ndx)}) | |
23 | 5, 22 | mp1i 13 | . . . . . 6 ⊢ (𝑉 ≠ (Base‘ndx) → dom {〈(Base‘ndx), 𝑉〉} = {(Base‘ndx)}) |
24 | 21, 23 | syl5eq 2794 | . . . . 5 ⊢ (𝑉 ≠ (Base‘ndx) → dom 𝐺 = {(Base‘ndx)}) |
25 | 20, 24 | neleqtrrd 2849 | . . . 4 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ (.ef‘ndx) ∈ dom 𝐺) |
26 | ndmfv 6367 | . . . 4 ⊢ (¬ (.ef‘ndx) ∈ dom 𝐺 → (𝐺‘(.ef‘ndx)) = ∅) | |
27 | 25, 26 | syl 17 | . . 3 ⊢ (𝑉 ≠ (Base‘ndx) → (𝐺‘(.ef‘ndx)) = ∅) |
28 | 14, 27 | syl5eq 2794 | . 2 ⊢ (𝑉 ≠ (Base‘ndx) → (.ef‘𝐺) = ∅) |
29 | 2, 9, 28 | 3eqtrd 2786 | 1 ⊢ (𝑉 ≠ (Base‘ndx) → (iEdg‘𝐺) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1620 ∈ wcel 2127 ≠ wne 2920 Vcvv 3328 ∅c0 4046 ifcif 4218 {csn 4309 〈cop 4315 × cxp 5252 dom cdm 5254 ‘cfv 6037 2nd c2nd 7320 ndxcnx 16027 Basecbs 16030 .efcedgf 26037 iEdgciedg 26045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-fal 1626 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-iun 4662 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-om 7219 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-er 7899 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-nn 11184 df-2 11242 df-3 11243 df-4 11244 df-5 11245 df-6 11246 df-7 11247 df-8 11248 df-9 11249 df-n0 11456 df-z 11541 df-dec 11657 df-ndx 16033 df-slot 16034 df-base 16036 df-edgf 26038 df-iedg 26047 |
This theorem is referenced by: (None) |
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