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Mirrors > Home > MPE Home > Th. List > mreexfidimd | Structured version Visualization version GIF version |
Description: In a Moore system whose closure operator has the exchange property, if two independent sets have equal closure and one is finite, then they are equinumerous. Proven by using mreexdomd 17602 twice. This implies a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mreexfidimd.1 | β’ (π β π΄ β (Mooreβπ)) |
mreexfidimd.2 | β’ π = (mrClsβπ΄) |
mreexfidimd.3 | β’ πΌ = (mrIndβπ΄) |
mreexfidimd.4 | β’ (π β βπ β π« πβπ¦ β π βπ§ β ((πβ(π βͺ {π¦})) β (πβπ ))π¦ β (πβ(π βͺ {π§}))) |
mreexfidimd.5 | β’ (π β π β πΌ) |
mreexfidimd.6 | β’ (π β π β πΌ) |
mreexfidimd.7 | β’ (π β π β Fin) |
mreexfidimd.8 | β’ (π β (πβπ) = (πβπ)) |
Ref | Expression |
---|---|
mreexfidimd | β’ (π β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mreexfidimd.1 | . . 3 β’ (π β π΄ β (Mooreβπ)) | |
2 | mreexfidimd.2 | . . 3 β’ π = (mrClsβπ΄) | |
3 | mreexfidimd.3 | . . 3 β’ πΌ = (mrIndβπ΄) | |
4 | mreexfidimd.4 | . . 3 β’ (π β βπ β π« πβπ¦ β π βπ§ β ((πβ(π βͺ {π¦})) β (πβπ ))π¦ β (πβ(π βͺ {π§}))) | |
5 | mreexfidimd.5 | . . . . . 6 β’ (π β π β πΌ) | |
6 | 3, 1, 5 | mrissd 17589 | . . . . 5 β’ (π β π β π) |
7 | 1, 2, 6 | mrcssidd 17578 | . . . 4 β’ (π β π β (πβπ)) |
8 | mreexfidimd.8 | . . . 4 β’ (π β (πβπ) = (πβπ)) | |
9 | 7, 8 | sseqtrd 4017 | . . 3 β’ (π β π β (πβπ)) |
10 | mreexfidimd.6 | . . . 4 β’ (π β π β πΌ) | |
11 | 3, 1, 10 | mrissd 17589 | . . 3 β’ (π β π β π) |
12 | mreexfidimd.7 | . . . 4 β’ (π β π β Fin) | |
13 | 12 | orcd 870 | . . 3 β’ (π β (π β Fin β¨ π β Fin)) |
14 | 1, 2, 3, 4, 9, 11, 13, 5 | mreexdomd 17602 | . 2 β’ (π β π βΌ π) |
15 | 1, 2, 11 | mrcssidd 17578 | . . . 4 β’ (π β π β (πβπ)) |
16 | 15, 8 | sseqtrrd 4018 | . . 3 β’ (π β π β (πβπ)) |
17 | 12 | olcd 871 | . . 3 β’ (π β (π β Fin β¨ π β Fin)) |
18 | 1, 2, 3, 4, 16, 6, 17, 10 | mreexdomd 17602 | . 2 β’ (π β π βΌ π) |
19 | sbth 9095 | . 2 β’ ((π βΌ π β§ π βΌ π) β π β π) | |
20 | 14, 18, 19 | syl2anc 583 | 1 β’ (π β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βwral 3055 β cdif 3940 βͺ cun 3941 π« cpw 4597 {csn 4623 class class class wbr 5141 βcfv 6537 β cen 8938 βΌ cdom 8939 Fincfn 8941 Moorecmre 17535 mrClscmrc 17536 mrIndcmri 17537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-om 7853 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-card 9936 df-mre 17539 df-mrc 17540 df-mri 17541 |
This theorem is referenced by: acsexdimd 18524 lvecdimfi 33200 |
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