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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 25934 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 25879 | . . 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 2847 | . . . 4 ⊢ (𝑉 ≠ (Base‘ndx) ↔ (Base‘ndx) ≠ 𝑉) | |
4 | fvex 6201 | . . . . 5 ⊢ (Base‘ndx) ∈ V | |
5 | snstrvtxval.v | . . . . 5 ⊢ 𝑉 ∈ V | |
6 | snstrvtxval.g | . . . . 5 ⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉} | |
7 | 4, 5, 6 | funsndifnop 6416 | . . . 4 ⊢ ((Base‘ndx) ≠ 𝑉 → ¬ 𝐺 ∈ (V × V)) |
8 | 3, 7 | sylbi 207 | . . 3 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ 𝐺 ∈ (V × V)) |
9 | 8 | iffalsed 4097 | . 2 ⊢ (𝑉 ≠ (Base‘ndx) → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (.ef‘𝐺)) |
10 | snex 4908 | . . . . . 6 ⊢ {〈(Base‘ndx), 𝑉〉} ∈ V | |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝐺 = {〈(Base‘ndx), 𝑉〉} → {〈(Base‘ndx), 𝑉〉} ∈ V) |
12 | 6, 11 | syl5eqel 2705 | . . . 4 ⊢ (𝐺 = {〈(Base‘ndx), 𝑉〉} → 𝐺 ∈ V) |
13 | edgfndxid 25871 | . . . 4 ⊢ (𝐺 ∈ V → (.ef‘𝐺) = (𝐺‘(.ef‘ndx))) | |
14 | 6, 12, 13 | mp2b 10 | . . 3 ⊢ (.ef‘𝐺) = (𝐺‘(.ef‘ndx)) |
15 | slotsbaseefdif 25873 | . . . . . . . 8 ⊢ (Base‘ndx) ≠ (.ef‘ndx) | |
16 | 15 | nesymi 2851 | . . . . . . 7 ⊢ ¬ (.ef‘ndx) = (Base‘ndx) |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ (.ef‘ndx) = (Base‘ndx)) |
18 | fvex 6201 | . . . . . . 7 ⊢ (.ef‘ndx) ∈ V | |
19 | 18 | elsn 4192 | . . . . . 6 ⊢ ((.ef‘ndx) ∈ {(Base‘ndx)} ↔ (.ef‘ndx) = (Base‘ndx)) |
20 | 17, 19 | sylnibr 319 | . . . . 5 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ (.ef‘ndx) ∈ {(Base‘ndx)}) |
21 | 6 | dmeqi 5325 | . . . . . 6 ⊢ dom 𝐺 = dom {〈(Base‘ndx), 𝑉〉} |
22 | dmsnopg 5606 | . . . . . . 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 2668 | . . . . 5 ⊢ (𝑉 ≠ (Base‘ndx) → dom 𝐺 = {(Base‘ndx)}) |
25 | 20, 24 | neleqtrrd 2723 | . . . 4 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ (.ef‘ndx) ∈ dom 𝐺) |
26 | ndmfv 6218 | . . . 4 ⊢ (¬ (.ef‘ndx) ∈ dom 𝐺 → (𝐺‘(.ef‘ndx)) = ∅) | |
27 | 25, 26 | syl 17 | . . 3 ⊢ (𝑉 ≠ (Base‘ndx) → (𝐺‘(.ef‘ndx)) = ∅) |
28 | 14, 27 | syl5eq 2668 | . 2 ⊢ (𝑉 ≠ (Base‘ndx) → (.ef‘𝐺) = ∅) |
29 | 2, 9, 28 | 3eqtrd 2660 | 1 ⊢ (𝑉 ≠ (Base‘ndx) → (iEdg‘𝐺) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∅c0 3915 ifcif 4086 {csn 4177 〈cop 4183 × cxp 5112 dom cdm 5114 ‘cfv 5888 2nd c2nd 7167 ndxcnx 15854 Basecbs 15857 .efcedgf 25867 iEdgciedg 25875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-ndx 15860 df-slot 15861 df-base 15863 df-edgf 25868 df-iedg 25877 |
This theorem is referenced by: (None) |
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