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Mirrors > Home > MPE Home > Th. List > structiedg0val | Structured version Visualization version GIF version |
Description: The set of indexed edges of an extensible structure with a base set and another slot not being the slot for edge functions is empty. (Contributed by AV, 23-Sep-2020.) (Proof shortened by AV, 12-Nov-2021.) |
Ref | Expression |
---|---|
structvtxvallem.s | ⊢ 𝑆 ∈ ℕ |
structvtxvallem.b | ⊢ (Base‘ndx) < 𝑆 |
structvtxvallem.g | ⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} |
Ref | Expression |
---|---|
structiedg0val | ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | structvtxvallem.g | . . . . 5 ⊢ 𝐺 = {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} | |
2 | structvtxvallem.b | . . . . 5 ⊢ (Base‘ndx) < 𝑆 | |
3 | structvtxvallem.s | . . . . 5 ⊢ 𝑆 ∈ ℕ | |
4 | 1, 2, 3 | 2strstr1 16597 | . . . 4 ⊢ 𝐺 Struct 〈(Base‘ndx), 𝑆〉 |
5 | structn0fun 16487 | . . . . 5 ⊢ (𝐺 Struct 〈(Base‘ndx), 𝑆〉 → Fun (𝐺 ∖ {∅})) | |
6 | 3, 2, 1 | structvtxvallem 26813 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 2 ≤ (♯‘dom 𝐺)) |
7 | funiedgdmge2val 26805 | . . . . 5 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺)) | |
8 | 5, 6, 7 | syl2an 598 | . . . 4 ⊢ ((𝐺 Struct 〈(Base‘ndx), 𝑆〉 ∧ (𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (iEdg‘𝐺) = (.ef‘𝐺)) |
9 | 4, 8 | mpan 689 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐺) = (.ef‘𝐺)) |
10 | 9 | 3adant3 1129 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = (.ef‘𝐺)) |
11 | prex 5298 | . . . . . 6 ⊢ {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} ∈ V | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝐺 = {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} → {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} ∈ V) |
13 | 1, 12 | eqeltrid 2894 | . . . 4 ⊢ (𝐺 = {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} → 𝐺 ∈ V) |
14 | edgfndxid 26786 | . . . 4 ⊢ (𝐺 ∈ V → (.ef‘𝐺) = (𝐺‘(.ef‘ndx))) | |
15 | 1, 13, 14 | mp2b 10 | . . 3 ⊢ (.ef‘𝐺) = (𝐺‘(.ef‘ndx)) |
16 | slotsbaseefdif 26788 | . . . . . . . . 9 ⊢ (Base‘ndx) ≠ (.ef‘ndx) | |
17 | 16 | nesymi 3044 | . . . . . . . 8 ⊢ ¬ (.ef‘ndx) = (Base‘ndx) |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) = (Base‘ndx)) |
19 | neneq 2993 | . . . . . . . . 9 ⊢ (𝑆 ≠ (.ef‘ndx) → ¬ 𝑆 = (.ef‘ndx)) | |
20 | eqcom 2805 | . . . . . . . . 9 ⊢ ((.ef‘ndx) = 𝑆 ↔ 𝑆 = (.ef‘ndx)) | |
21 | 19, 20 | sylnibr 332 | . . . . . . . 8 ⊢ (𝑆 ≠ (.ef‘ndx) → ¬ (.ef‘ndx) = 𝑆) |
22 | 21 | 3ad2ant3 1132 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) = 𝑆) |
23 | ioran 981 | . . . . . . 7 ⊢ (¬ ((.ef‘ndx) = (Base‘ndx) ∨ (.ef‘ndx) = 𝑆) ↔ (¬ (.ef‘ndx) = (Base‘ndx) ∧ ¬ (.ef‘ndx) = 𝑆)) | |
24 | 18, 22, 23 | sylanbrc 586 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ ((.ef‘ndx) = (Base‘ndx) ∨ (.ef‘ndx) = 𝑆)) |
25 | fvex 6658 | . . . . . . 7 ⊢ (.ef‘ndx) ∈ V | |
26 | 25 | elpr 4548 | . . . . . 6 ⊢ ((.ef‘ndx) ∈ {(Base‘ndx), 𝑆} ↔ ((.ef‘ndx) = (Base‘ndx) ∨ (.ef‘ndx) = 𝑆)) |
27 | 24, 26 | sylnibr 332 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) ∈ {(Base‘ndx), 𝑆}) |
28 | 1 | dmeqi 5737 | . . . . . 6 ⊢ dom 𝐺 = dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} |
29 | dmpropg 6039 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} = {(Base‘ndx), 𝑆}) | |
30 | 29 | 3adant3 1129 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} = {(Base‘ndx), 𝑆}) |
31 | 28, 30 | syl5eq 2845 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → dom 𝐺 = {(Base‘ndx), 𝑆}) |
32 | 27, 31 | neleqtrrd 2912 | . . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) ∈ dom 𝐺) |
33 | ndmfv 6675 | . . . 4 ⊢ (¬ (.ef‘ndx) ∈ dom 𝐺 → (𝐺‘(.ef‘ndx)) = ∅) | |
34 | 32, 33 | syl 17 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (𝐺‘(.ef‘ndx)) = ∅) |
35 | 15, 34 | syl5eq 2845 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (.ef‘𝐺) = ∅) |
36 | 10, 35 | eqtrd 2833 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 844 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 Vcvv 3441 ∖ cdif 3878 ∅c0 4243 {csn 4525 {cpr 4527 〈cop 4531 class class class wbr 5030 dom cdm 5519 Fun wfun 6318 ‘cfv 6324 < clt 10664 ≤ cle 10665 ℕcn 11625 2c2 11680 ♯chash 13686 Struct cstr 16471 ndxcnx 16472 Basecbs 16475 .efcedgf 26782 iEdgciedg 26790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-hash 13687 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-edgf 26783 df-iedg 26792 |
This theorem is referenced by: (None) |
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