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Mirrors > Home > MPE Home > Th. List > psrbagconcl | Structured version Visualization version GIF version |
Description: The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 6-Aug-2024.) |
Ref | Expression |
---|---|
psrbag.d | β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} |
psrbagconf1o.s | β’ π = {π¦ β π· β£ π¦ βr β€ πΉ} |
Ref | Expression |
---|---|
psrbagconcl | β’ ((πΉ β π· β§ π β π) β (πΉ βf β π) β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 481 | . . 3 β’ ((πΉ β π· β§ π β π) β πΉ β π·) | |
2 | simpr 483 | . . . . . 6 β’ ((πΉ β π· β§ π β π) β π β π) | |
3 | breq1 5147 | . . . . . . 7 β’ (π¦ = π β (π¦ βr β€ πΉ β π βr β€ πΉ)) | |
4 | psrbagconf1o.s | . . . . . . 7 β’ π = {π¦ β π· β£ π¦ βr β€ πΉ} | |
5 | 3, 4 | elrab2 3679 | . . . . . 6 β’ (π β π β (π β π· β§ π βr β€ πΉ)) |
6 | 2, 5 | sylib 217 | . . . . 5 β’ ((πΉ β π· β§ π β π) β (π β π· β§ π βr β€ πΉ)) |
7 | 6 | simpld 493 | . . . 4 β’ ((πΉ β π· β§ π β π) β π β π·) |
8 | psrbag.d | . . . . 5 β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} | |
9 | 8 | psrbagf 21850 | . . . 4 β’ (π β π· β π:πΌβΆβ0) |
10 | 7, 9 | syl 17 | . . 3 β’ ((πΉ β π· β§ π β π) β π:πΌβΆβ0) |
11 | 6 | simprd 494 | . . 3 β’ ((πΉ β π· β§ π β π) β π βr β€ πΉ) |
12 | 8 | psrbagcon 21862 | . . 3 β’ ((πΉ β π· β§ π:πΌβΆβ0 β§ π βr β€ πΉ) β ((πΉ βf β π) β π· β§ (πΉ βf β π) βr β€ πΉ)) |
13 | 1, 10, 11, 12 | syl3anc 1368 | . 2 β’ ((πΉ β π· β§ π β π) β ((πΉ βf β π) β π· β§ (πΉ βf β π) βr β€ πΉ)) |
14 | breq1 5147 | . . 3 β’ (π¦ = (πΉ βf β π) β (π¦ βr β€ πΉ β (πΉ βf β π) βr β€ πΉ)) | |
15 | 14, 4 | elrab2 3679 | . 2 β’ ((πΉ βf β π) β π β ((πΉ βf β π) β π· β§ (πΉ βf β π) βr β€ πΉ)) |
16 | 13, 15 | sylibr 233 | 1 β’ ((πΉ β π· β§ π β π) β (πΉ βf β π) β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 {crab 3419 class class class wbr 5144 β‘ccnv 5672 β cima 5676 βΆwf 6539 (class class class)co 7413 βf cof 7677 βr cofr 7678 βm cmap 8838 Fincfn 8957 β€ cle 11274 β cmin 11469 βcn 12237 β0cn0 12497 |
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 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-ofr 7680 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 |
This theorem is referenced by: psrbagconf1o 21869 psrass1lem 21876 psrdi 21909 psrdir 21910 psrass23l 21911 psrcom 21912 psrass23 21913 resspsrmul 21920 mplsubrglem 21948 mplmonmul 21976 mhpmulcl 22076 psdmul 22093 psropprmul 22160 mdegmullem 26027 rhmcomulpsr 41835 rhmcomulmpl 41839 |
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