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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 16605 | . . . 4 ⊢ 𝐺 Struct 〈(Base‘ndx), 𝑆〉 |
5 | structn0fun 16495 | . . . . 5 ⊢ (𝐺 Struct 〈(Base‘ndx), 𝑆〉 → Fun (𝐺 ∖ {∅})) | |
6 | 3, 2, 1 | structvtxvallem 26805 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 2 ≤ (♯‘dom 𝐺)) |
7 | funiedgdmge2val 26797 | . . . . 5 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺)) | |
8 | 5, 6, 7 | syl2an 597 | . . . 4 ⊢ ((𝐺 Struct 〈(Base‘ndx), 𝑆〉 ∧ (𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (iEdg‘𝐺) = (.ef‘𝐺)) |
9 | 4, 8 | mpan 688 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐺) = (.ef‘𝐺)) |
10 | 9 | 3adant3 1128 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = (.ef‘𝐺)) |
11 | prex 5333 | . . . . . 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 2917 | . . . 4 ⊢ (𝐺 = {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} → 𝐺 ∈ V) |
14 | edgfndxid 26778 | . . . 4 ⊢ (𝐺 ∈ V → (.ef‘𝐺) = (𝐺‘(.ef‘ndx))) | |
15 | 1, 13, 14 | mp2b 10 | . . 3 ⊢ (.ef‘𝐺) = (𝐺‘(.ef‘ndx)) |
16 | slotsbaseefdif 26780 | . . . . . . . . 9 ⊢ (Base‘ndx) ≠ (.ef‘ndx) | |
17 | 16 | nesymi 3073 | . . . . . . . 8 ⊢ ¬ (.ef‘ndx) = (Base‘ndx) |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) = (Base‘ndx)) |
19 | neneq 3022 | . . . . . . . . 9 ⊢ (𝑆 ≠ (.ef‘ndx) → ¬ 𝑆 = (.ef‘ndx)) | |
20 | eqcom 2828 | . . . . . . . . 9 ⊢ ((.ef‘ndx) = 𝑆 ↔ 𝑆 = (.ef‘ndx)) | |
21 | 19, 20 | sylnibr 331 | . . . . . . . 8 ⊢ (𝑆 ≠ (.ef‘ndx) → ¬ (.ef‘ndx) = 𝑆) |
22 | 21 | 3ad2ant3 1131 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) = 𝑆) |
23 | ioran 980 | . . . . . . 7 ⊢ (¬ ((.ef‘ndx) = (Base‘ndx) ∨ (.ef‘ndx) = 𝑆) ↔ (¬ (.ef‘ndx) = (Base‘ndx) ∧ ¬ (.ef‘ndx) = 𝑆)) | |
24 | 18, 22, 23 | sylanbrc 585 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ ((.ef‘ndx) = (Base‘ndx) ∨ (.ef‘ndx) = 𝑆)) |
25 | fvex 6683 | . . . . . . 7 ⊢ (.ef‘ndx) ∈ V | |
26 | 25 | elpr 4590 | . . . . . 6 ⊢ ((.ef‘ndx) ∈ {(Base‘ndx), 𝑆} ↔ ((.ef‘ndx) = (Base‘ndx) ∨ (.ef‘ndx) = 𝑆)) |
27 | 24, 26 | sylnibr 331 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) ∈ {(Base‘ndx), 𝑆}) |
28 | 1 | dmeqi 5773 | . . . . . 6 ⊢ dom 𝐺 = dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} |
29 | dmpropg 6072 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} = {(Base‘ndx), 𝑆}) | |
30 | 29 | 3adant3 1128 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} = {(Base‘ndx), 𝑆}) |
31 | 28, 30 | syl5eq 2868 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → dom 𝐺 = {(Base‘ndx), 𝑆}) |
32 | 27, 31 | neleqtrrd 2935 | . . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) ∈ dom 𝐺) |
33 | ndmfv 6700 | . . . 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 2868 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (.ef‘𝐺) = ∅) |
36 | 10, 35 | eqtrd 2856 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ∖ cdif 3933 ∅c0 4291 {csn 4567 {cpr 4569 〈cop 4573 class class class wbr 5066 dom cdm 5555 Fun wfun 6349 ‘cfv 6355 < clt 10675 ≤ cle 10676 ℕcn 11638 2c2 11693 ♯chash 13691 Struct cstr 16479 ndxcnx 16480 Basecbs 16483 .efcedgf 26774 iEdgciedg 26782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-edgf 26775 df-iedg 26784 |
This theorem is referenced by: (None) |
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