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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 484 | . . 3 β’ ((πΉ β π· β§ π β π) β πΉ β π·) | |
2 | simpr 486 | . . . . . 6 β’ ((πΉ β π· β§ π β π) β π β π) | |
3 | breq1 5109 | . . . . . . 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 3649 | . . . . . 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 496 | . . . 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 21336 | . . . 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 497 | . . 3 β’ ((πΉ β π· β§ π β π) β π βr β€ πΉ) |
12 | 8 | psrbagcon 21348 | . . 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 1372 | . 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 5109 | . . 3 β’ (π¦ = (πΉ βf β π) β (π¦ βr β€ πΉ β (πΉ βf β π) βr β€ πΉ)) | |
15 | 14, 4 | elrab2 3649 | . 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 397 = wceq 1542 β wcel 2107 {crab 3406 class class class wbr 5106 β‘ccnv 5633 β cima 5637 βΆwf 6493 (class class class)co 7358 βf cof 7616 βr cofr 7617 βm cmap 8768 Fincfn 8886 β€ cle 11195 β cmin 11390 βcn 12158 β0cn0 12418 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-ofr 7619 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 |
This theorem is referenced by: psrbagconf1o 21354 psrass1lem 21361 psrdi 21391 psrdir 21392 psrass23l 21393 psrcom 21394 psrass23 21395 resspsrmul 21402 mplsubrglem 21426 mplmonmul 21453 mhpmulcl 21555 psropprmul 21625 mdegmullem 25459 |
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