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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 16710 | . . . 4 ⊢ 𝐺 Struct 〈(Base‘ndx), 𝑆〉 |
5 | structn0fun 16600 | . . . . 5 ⊢ (𝐺 Struct 〈(Base‘ndx), 𝑆〉 → Fun (𝐺 ∖ {∅})) | |
6 | 3, 2, 1 | structvtxvallem 26967 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 2 ≤ (♯‘dom 𝐺)) |
7 | funiedgdmge2val 26959 | . . . . 5 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 2 ≤ (♯‘dom 𝐺)) → (iEdg‘𝐺) = (.ef‘𝐺)) | |
8 | 5, 6, 7 | syl2an 599 | . . . 4 ⊢ ((𝐺 Struct 〈(Base‘ndx), 𝑆〉 ∧ (𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (iEdg‘𝐺) = (.ef‘𝐺)) |
9 | 4, 8 | mpan 690 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘𝐺) = (.ef‘𝐺)) |
10 | 9 | 3adant3 1133 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (iEdg‘𝐺) = (.ef‘𝐺)) |
11 | prex 5299 | . . . . . 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 2837 | . . . 4 ⊢ (𝐺 = {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} → 𝐺 ∈ V) |
14 | edgfndxid 26940 | . . . 4 ⊢ (𝐺 ∈ V → (.ef‘𝐺) = (𝐺‘(.ef‘ndx))) | |
15 | 1, 13, 14 | mp2b 10 | . . 3 ⊢ (.ef‘𝐺) = (𝐺‘(.ef‘ndx)) |
16 | slotsbaseefdif 26942 | . . . . . . . . 9 ⊢ (Base‘ndx) ≠ (.ef‘ndx) | |
17 | 16 | nesymi 2991 | . . . . . . . 8 ⊢ ¬ (.ef‘ndx) = (Base‘ndx) |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) = (Base‘ndx)) |
19 | neneq 2940 | . . . . . . . . 9 ⊢ (𝑆 ≠ (.ef‘ndx) → ¬ 𝑆 = (.ef‘ndx)) | |
20 | eqcom 2745 | . . . . . . . . 9 ⊢ ((.ef‘ndx) = 𝑆 ↔ 𝑆 = (.ef‘ndx)) | |
21 | 19, 20 | sylnibr 332 | . . . . . . . 8 ⊢ (𝑆 ≠ (.ef‘ndx) → ¬ (.ef‘ndx) = 𝑆) |
22 | 21 | 3ad2ant3 1136 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) = 𝑆) |
23 | ioran 983 | . . . . . . 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 6689 | . . . . . . 7 ⊢ (.ef‘ndx) ∈ V | |
26 | 25 | elpr 4539 | . . . . . 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 5747 | . . . . . 6 ⊢ dom 𝐺 = dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} |
29 | dmpropg 6047 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} = {(Base‘ndx), 𝑆}) | |
30 | 29 | 3adant3 1133 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → dom {〈(Base‘ndx), 𝑉〉, 〈𝑆, 𝐸〉} = {(Base‘ndx), 𝑆}) |
31 | 28, 30 | syl5eq 2785 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → dom 𝐺 = {(Base‘ndx), 𝑆}) |
32 | 27, 31 | neleqtrrd 2855 | . . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → ¬ (.ef‘ndx) ∈ dom 𝐺) |
33 | ndmfv 6706 | . . . 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 2785 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝑆 ≠ (.ef‘ndx)) → (.ef‘𝐺) = ∅) |
36 | 10, 35 | eqtrd 2773 | 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 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 Vcvv 3398 ∖ cdif 3840 ∅c0 4211 {csn 4516 {cpr 4518 〈cop 4522 class class class wbr 5030 dom cdm 5525 Fun wfun 6333 ‘cfv 6339 < clt 10755 ≤ cle 10756 ℕcn 11718 2c2 11773 ♯chash 13784 Struct cstr 16584 ndxcnx 16585 Basecbs 16588 .efcedgf 26936 iEdgciedg 26944 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-oadd 8137 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-dju 9405 df-card 9443 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-7 11786 df-8 11787 df-9 11788 df-n0 11979 df-xnn0 12051 df-z 12065 df-dec 12182 df-uz 12327 df-fz 12984 df-hash 13785 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-edgf 26937 df-iedg 26946 |
This theorem is referenced by: (None) |
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