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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 26830 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 26789 | . . 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 3072 | . . . 4 ⊢ (𝑉 ≠ (Base‘ndx) ↔ (Base‘ndx) ≠ 𝑉) | |
4 | fvex 6686 | . . . . 5 ⊢ (Base‘ndx) ∈ V | |
5 | snstrvtxval.v | . . . . 5 ⊢ 𝑉 ∈ V | |
6 | snstrvtxval.g | . . . . 5 ⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉} | |
7 | 4, 5, 6 | funsndifnop 6916 | . . . 4 ⊢ ((Base‘ndx) ≠ 𝑉 → ¬ 𝐺 ∈ (V × V)) |
8 | 3, 7 | sylbi 219 | . . 3 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ 𝐺 ∈ (V × V)) |
9 | 8 | iffalsed 4481 | . 2 ⊢ (𝑉 ≠ (Base‘ndx) → if(𝐺 ∈ (V × V), (2nd ‘𝐺), (.ef‘𝐺)) = (.ef‘𝐺)) |
10 | snex 5335 | . . . . . 6 ⊢ {〈(Base‘ndx), 𝑉〉} ∈ V | |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝐺 = {〈(Base‘ndx), 𝑉〉} → {〈(Base‘ndx), 𝑉〉} ∈ V) |
12 | 6, 11 | eqeltrid 2920 | . . . 4 ⊢ (𝐺 = {〈(Base‘ndx), 𝑉〉} → 𝐺 ∈ V) |
13 | edgfndxid 26781 | . . . 4 ⊢ (𝐺 ∈ V → (.ef‘𝐺) = (𝐺‘(.ef‘ndx))) | |
14 | 6, 12, 13 | mp2b 10 | . . 3 ⊢ (.ef‘𝐺) = (𝐺‘(.ef‘ndx)) |
15 | slotsbaseefdif 26783 | . . . . . . . 8 ⊢ (Base‘ndx) ≠ (.ef‘ndx) | |
16 | 15 | nesymi 3076 | . . . . . . 7 ⊢ ¬ (.ef‘ndx) = (Base‘ndx) |
17 | 16 | a1i 11 | . . . . . 6 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ (.ef‘ndx) = (Base‘ndx)) |
18 | fvex 6686 | . . . . . . 7 ⊢ (.ef‘ndx) ∈ V | |
19 | 18 | elsn 4585 | . . . . . 6 ⊢ ((.ef‘ndx) ∈ {(Base‘ndx)} ↔ (.ef‘ndx) = (Base‘ndx)) |
20 | 17, 19 | sylnibr 331 | . . . . 5 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ (.ef‘ndx) ∈ {(Base‘ndx)}) |
21 | 6 | dmeqi 5776 | . . . . . 6 ⊢ dom 𝐺 = dom {〈(Base‘ndx), 𝑉〉} |
22 | dmsnopg 6073 | . . . . . . 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 2871 | . . . . 5 ⊢ (𝑉 ≠ (Base‘ndx) → dom 𝐺 = {(Base‘ndx)}) |
25 | 20, 24 | neleqtrrd 2938 | . . . 4 ⊢ (𝑉 ≠ (Base‘ndx) → ¬ (.ef‘ndx) ∈ dom 𝐺) |
26 | ndmfv 6703 | . . . 4 ⊢ (¬ (.ef‘ndx) ∈ dom 𝐺 → (𝐺‘(.ef‘ndx)) = ∅) | |
27 | 25, 26 | syl 17 | . . 3 ⊢ (𝑉 ≠ (Base‘ndx) → (𝐺‘(.ef‘ndx)) = ∅) |
28 | 14, 27 | syl5eq 2871 | . 2 ⊢ (𝑉 ≠ (Base‘ndx) → (.ef‘𝐺) = ∅) |
29 | 2, 9, 28 | 3eqtrd 2863 | 1 ⊢ (𝑉 ≠ (Base‘ndx) → (iEdg‘𝐺) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 Vcvv 3497 ∅c0 4294 ifcif 4470 {csn 4570 〈cop 4576 × cxp 5556 dom cdm 5558 ‘cfv 6358 2nd c2nd 7691 ndxcnx 16483 Basecbs 16486 .efcedgf 26777 iEdgciedg 26785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-ndx 16489 df-slot 16490 df-base 16492 df-edgf 26778 df-iedg 26787 |
This theorem is referenced by: (None) |
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