![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 28829 for the (degenerate) 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 28788 | . . 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 2989 | . . . 4 ⊢ (𝑉 ≠ (Base‘ndx) ↔ (Base‘ndx) ≠ 𝑉) | |
4 | fvex 6904 | . . . . 5 ⊢ (Base‘ndx) ∈ V | |
5 | snstrvtxval.v | . . . . 5 ⊢ 𝑉 ∈ V | |
6 | snstrvtxval.g | . . . . 5 ⊢ 𝐺 = {⟨(Base‘ndx), 𝑉⟩} | |
7 | 4, 5, 6 | funsndifnop 7154 | . . . 4 ⊢ ((Base‘ndx) ≠ 𝑉 → ¬ 𝐺 ∈ (V × V)) |
8 | 3, 7 | sylbi 216 | . . 3 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ 𝐺 ∈ (V × V)) |
9 | 8 | iffalsed 4535 | . 2 ⊢ (𝑉 ≠ (Base‘ndx) → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (.ef‘𝐺)) |
10 | snex 5427 | . . . . . 6 ⊢ {⟨(Base‘ndx), 𝑉⟩} ∈ V | |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝐺 = {⟨(Base‘ndx), 𝑉⟩} → {⟨(Base‘ndx), 𝑉⟩} ∈ V) |
12 | 6, 11 | eqeltrid 2832 | . . . 4 ⊢ (𝐺 = {⟨(Base‘ndx), 𝑉⟩} → 𝐺 ∈ V) |
13 | edgfndxid 28778 | . . . 4 ⊢ (𝐺 ∈ V → (.ef‘𝐺) = (𝐺‘(.ef‘ndx))) | |
14 | 6, 12, 13 | mp2b 10 | . . 3 ⊢ (.ef‘𝐺) = (𝐺‘(.ef‘ndx)) |
15 | basendxnedgfndx 28782 | . . . . . . . 8 ⊢ (Base‘ndx) ≠ (.ef‘ndx) | |
16 | 15 | nesymi 2993 | . . . . . . 7 ⊢ ¬ (.ef‘ndx) = (Base‘ndx) |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ (.ef‘ndx) = (Base‘ndx)) |
18 | fvex 6904 | . . . . . . 7 ⊢ (.ef‘ndx) ∈ V | |
19 | 18 | elsn 4639 | . . . . . 6 ⊢ ((.ef‘ndx) ∈ {(Base‘ndx)} ↔ (.ef‘ndx) = (Base‘ndx)) |
20 | 17, 19 | sylnibr 329 | . . . . 5 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ (.ef‘ndx) ∈ {(Base‘ndx)}) |
21 | 6 | dmeqi 5901 | . . . . . 6 ⊢ dom 𝐺 = dom {⟨(Base‘ndx), 𝑉⟩} |
22 | dmsnopg 6211 | . . . . . . 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 | eqtrid 2779 | . . . . 5 ⊢ (𝑉 ≠ (Base‘ndx) → dom 𝐺 = {(Base‘ndx)}) |
25 | 20, 24 | neleqtrrd 2851 | . . . 4 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ (.ef‘ndx) ∈ dom 𝐺) |
26 | ndmfv 6926 | . . . 4 ⊢ (¬ (.ef‘ndx) ∈ dom 𝐺 → (𝐺‘(.ef‘ndx)) = ∅) | |
27 | 25, 26 | syl 17 | . . 3 ⊢ (𝑉 ≠ (Base‘ndx) → (𝐺‘(.ef‘ndx)) = ∅) |
28 | 14, 27 | eqtrid 2779 | . 2 ⊢ (𝑉 ≠ (Base‘ndx) → (.ef‘𝐺) = ∅) |
29 | 2, 9, 28 | 3eqtrd 2771 | 1 ⊢ (𝑉 ≠ (Base‘ndx) → (iEdg‘𝐺) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 Vcvv 3469 ∅c0 4318 ifcif 4524 {csn 4624 ⟨cop 4630 × cxp 5670 dom cdm 5672 ‘cfv 6542 2nd c2nd 7984 ndxcnx 17147 Basecbs 17165 .efcedgf 28773 iEdgciedg 28784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-slot 17136 df-ndx 17148 df-base 17166 df-edgf 28774 df-iedg 28786 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |