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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 17278 | . . . 4 ⊢ 𝐺 Struct 〈(Base‘ndx), 𝑆〉 |
5 | structn0fun 17193 | . . . . 5 ⊢ (𝐺 Struct 〈(Base‘ndx), 𝑆〉 → Fun (𝐺 ∖ {∅})) | |
6 | 3, 2, 1 | structvtxvallem 29046 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 2 ≤ (♯‘dom 𝐺)) |
7 | funiedgdmge2val 29038 | . . . . 5 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺)) | |
8 | 5, 6, 7 | syl2an 595 | . . . 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 1132 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = (.ef‘𝐺)) |
11 | prex 5455 | . . . . . 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 2842 | . . . 4 ⊢ (𝐺 = {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} → 𝐺 ∈ V) |
14 | edgfndxid 29017 | . . . 4 ⊢ (𝐺 ∈ V → (.ef‘𝐺) = (𝐺‘(.ef‘ndx))) | |
15 | 1, 13, 14 | mp2b 10 | . . 3 ⊢ (.ef‘𝐺) = (𝐺‘(.ef‘ndx)) |
16 | basendxnedgfndx 29021 | . . . . . . . . 9 ⊢ (Base‘ndx) ≠ (.ef‘ndx) | |
17 | 16 | nesymi 3000 | . . . . . . . 8 ⊢ ¬ (.ef‘ndx) = (Base‘ndx) |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) = (Base‘ndx)) |
19 | neneq 2948 | . . . . . . . . 9 ⊢ (𝑆 ≠ (.ef‘ndx) → ¬ 𝑆 = (.ef‘ndx)) | |
20 | eqcom 2741 | . . . . . . . . 9 ⊢ ((.ef‘ndx) = 𝑆 ↔ 𝑆 = (.ef‘ndx)) | |
21 | 19, 20 | sylnibr 329 | . . . . . . . 8 ⊢ (𝑆 ≠ (.ef‘ndx) → ¬ (.ef‘ndx) = 𝑆) |
22 | 21 | 3ad2ant3 1135 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) = 𝑆) |
23 | ioran 984 | . . . . . . 7 ⊢ (¬ ((.ef‘ndx) = (Base‘ndx) ∨ (.ef‘ndx) = 𝑆) ↔ (¬ (.ef‘ndx) = (Base‘ndx) ∧ ¬ (.ef‘ndx) = 𝑆)) | |
24 | 18, 22, 23 | sylanbrc 582 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ ((.ef‘ndx) = (Base‘ndx) ∨ (.ef‘ndx) = 𝑆)) |
25 | fvex 6932 | . . . . . . 7 ⊢ (.ef‘ndx) ∈ V | |
26 | 25 | elpr 4672 | . . . . . 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 5928 | . . . . . 6 ⊢ dom 𝐺 = dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} |
29 | dmpropg 6245 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} = {(Base‘ndx), 𝑆}) | |
30 | 29 | 3adant3 1132 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} = {(Base‘ndx), 𝑆}) |
31 | 28, 30 | eqtrid 2786 | . . . . 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 6954 | . . . 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 2786 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (.ef‘𝐺) = ∅) |
36 | 10, 35 | eqtrd 2774 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 846 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 ≠ wne 2942 Vcvv 3482 ∖ cdif 3967 ∅c0 4347 {csn 4648 {cpr 4650 〈cop 4654 class class class wbr 5169 dom cdm 5699 Fun wfun 6566 ‘cfv 6572 < clt 11320 ≤ cle 11321 ℕcn 12289 2c2 12344 ♯chash 14375 Struct cstr 17188 ndxcnx 17235 Basecbs 17253 .efcedgf 29012 iEdgciedg 29023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-oadd 8522 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-dju 9966 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-xnn0 12622 df-z 12636 df-dec 12755 df-uz 12900 df-fz 13564 df-hash 14376 df-struct 17189 df-slot 17224 df-ndx 17236 df-base 17254 df-edgf 29013 df-iedg 29025 |
This theorem is referenced by: (None) |
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