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Mirrors > Home > MPE Home > Th. List > acsficl | Structured version Visualization version GIF version |
Description: A closure in an algebraic closure system is the union of the closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.) |
Ref | Expression |
---|---|
acsdrscl.f | ⊢ 𝐹 = (mrCls‘𝐶) |
Ref | Expression |
---|---|
acsficl | ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝐹‘𝑆) = ∪ (𝐹 “ (𝒫 𝑆 ∩ Fin))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6220 | . . . 4 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ dom ACS) | |
2 | elpw2g 4827 | . . . 4 ⊢ (𝑋 ∈ dom ACS → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋)) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋)) |
4 | 3 | biimpar 502 | . 2 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋) |
5 | isacs3lem 17166 | . . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶))) | |
6 | acsdrscl.f | . . . . . 6 ⊢ 𝐹 = (mrCls‘𝐶) | |
7 | 6 | isacs4lem 17168 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑡 ∈ 𝒫 𝒫 𝑋((toInc‘𝑡) ∈ Dirset → (𝐹‘∪ 𝑡) = ∪ (𝐹 “ 𝑡)))) |
8 | 6 | isacs5lem 17169 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑡 ∈ 𝒫 𝒫 𝑋((toInc‘𝑡) ∈ Dirset → (𝐹‘∪ 𝑡) = ∪ (𝐹 “ 𝑡))) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝐹‘𝑠) = ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)))) |
9 | 5, 7, 8 | 3syl 18 | . . . 4 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝐹‘𝑠) = ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)))) |
10 | 9 | simprd 479 | . . 3 ⊢ (𝐶 ∈ (ACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝐹‘𝑠) = ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin))) |
11 | 10 | adantr 481 | . 2 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝐹‘𝑠) = ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin))) |
12 | fveq2 6191 | . . . 4 ⊢ (𝑠 = 𝑆 → (𝐹‘𝑠) = (𝐹‘𝑆)) | |
13 | pweq 4161 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆) | |
14 | 13 | ineq1d 3813 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑆 ∩ Fin)) |
15 | 14 | imaeq2d 5466 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑆 ∩ Fin))) |
16 | 15 | unieqd 4446 | . . . 4 ⊢ (𝑠 = 𝑆 → ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) = ∪ (𝐹 “ (𝒫 𝑆 ∩ Fin))) |
17 | 12, 16 | eqeq12d 2637 | . . 3 ⊢ (𝑠 = 𝑆 → ((𝐹‘𝑠) = ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin)) ↔ (𝐹‘𝑆) = ∪ (𝐹 “ (𝒫 𝑆 ∩ Fin)))) |
18 | 17 | rspcva 3307 | . 2 ⊢ ((𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝐹‘𝑠) = ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin))) → (𝐹‘𝑆) = ∪ (𝐹 “ (𝒫 𝑆 ∩ Fin))) |
19 | 4, 11, 18 | syl2anc 693 | 1 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝐹‘𝑆) = ∪ (𝐹 “ (𝒫 𝑆 ∩ Fin))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 dom cdm 5114 “ cima 5117 ‘cfv 5888 Fincfn 7955 Moorecmre 16242 mrClscmrc 16243 ACScacs 16245 Dirsetcdrs 16927 toInccipo 17151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-tset 15960 df-ple 15961 df-ocomp 15963 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-drs 16929 df-poset 16946 df-ipo 17152 |
This theorem is referenced by: acsficld 17175 |
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