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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 16937 | . . . 4 ⊢ 𝐺 Struct 〈(Base‘ndx), 𝑆〉 |
5 | structn0fun 16852 | . . . . 5 ⊢ (𝐺 Struct 〈(Base‘ndx), 𝑆〉 → Fun (𝐺 ∖ {∅})) | |
6 | 3, 2, 1 | structvtxvallem 27390 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 2 ≤ (♯‘dom 𝐺)) |
7 | funiedgdmge2val 27382 | . . . . 5 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺)) | |
8 | 5, 6, 7 | syl2an 596 | . . . 4 ⊢ ((𝐺 Struct 〈(Base‘ndx), 𝑆〉 ∧ (𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (iEdg‘𝐺) = (.ef‘𝐺)) |
9 | 4, 8 | mpan 687 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐺) = (.ef‘𝐺)) |
10 | 9 | 3adant3 1131 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = (.ef‘𝐺)) |
11 | prex 5355 | . . . . . 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 2843 | . . . 4 ⊢ (𝐺 = {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} → 𝐺 ∈ V) |
14 | edgfndxid 27361 | . . . 4 ⊢ (𝐺 ∈ V → (.ef‘𝐺) = (𝐺‘(.ef‘ndx))) | |
15 | 1, 13, 14 | mp2b 10 | . . 3 ⊢ (.ef‘𝐺) = (𝐺‘(.ef‘ndx)) |
16 | basendxnedgfndx 27365 | . . . . . . . . 9 ⊢ (Base‘ndx) ≠ (.ef‘ndx) | |
17 | 16 | nesymi 3001 | . . . . . . . 8 ⊢ ¬ (.ef‘ndx) = (Base‘ndx) |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) = (Base‘ndx)) |
19 | neneq 2949 | . . . . . . . . 9 ⊢ (𝑆 ≠ (.ef‘ndx) → ¬ 𝑆 = (.ef‘ndx)) | |
20 | eqcom 2745 | . . . . . . . . 9 ⊢ ((.ef‘ndx) = 𝑆 ↔ 𝑆 = (.ef‘ndx)) | |
21 | 19, 20 | sylnibr 329 | . . . . . . . 8 ⊢ (𝑆 ≠ (.ef‘ndx) → ¬ (.ef‘ndx) = 𝑆) |
22 | 21 | 3ad2ant3 1134 | . . . . . . 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 583 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ ((.ef‘ndx) = (Base‘ndx) ∨ (.ef‘ndx) = 𝑆)) |
25 | fvex 6787 | . . . . . . 7 ⊢ (.ef‘ndx) ∈ V | |
26 | 25 | elpr 4584 | . . . . . 6 ⊢ ((.ef‘ndx) ∈ {(Base‘ndx), 𝑆} ↔ ((.ef‘ndx) = (Base‘ndx) ∨ (.ef‘ndx) = 𝑆)) |
27 | 24, 26 | sylnibr 329 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) ∈ {(Base‘ndx), 𝑆}) |
28 | 1 | dmeqi 5813 | . . . . . 6 ⊢ dom 𝐺 = dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} |
29 | dmpropg 6118 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} = {(Base‘ndx), 𝑆}) | |
30 | 29 | 3adant3 1131 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} = {(Base‘ndx), 𝑆}) |
31 | 28, 30 | eqtrid 2790 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → dom 𝐺 = {(Base‘ndx), 𝑆}) |
32 | 27, 31 | neleqtrrd 2861 | . . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) ∈ dom 𝐺) |
33 | ndmfv 6804 | . . . 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 | eqtrid 2790 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (.ef‘𝐺) = ∅) |
36 | 10, 35 | eqtrd 2778 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 Vcvv 3432 ∖ cdif 3884 ∅c0 4256 {csn 4561 {cpr 4563 〈cop 4567 class class class wbr 5074 dom cdm 5589 Fun wfun 6427 ‘cfv 6433 < clt 11009 ≤ cle 11010 ℕcn 11973 2c2 12028 ♯chash 14044 Struct cstr 16847 ndxcnx 16894 Basecbs 16912 .efcedgf 27356 iEdgciedg 27367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-hash 14045 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-edgf 27357 df-iedg 27369 |
This theorem is referenced by: (None) |
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