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Mirrors > Home > MPE Home > Th. List > psrbagfsupp | Structured version Visualization version GIF version |
Description: Finite bags have finite support. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.) |
Ref | Expression |
---|---|
psrbag.d | β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} |
Ref | Expression |
---|---|
psrbagfsupp | β’ (πΉ β π· β πΉ finSupp 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . 5 β’ (πΉ β π· β πΉ β π·) | |
2 | psrbag.d | . . . . . . 7 β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} | |
3 | 2 | psrbagf 21336 | . . . . . 6 β’ (πΉ β π· β πΉ:πΌβΆβ0) |
4 | 3 | ffnd 6670 | . . . . 5 β’ (πΉ β π· β πΉ Fn πΌ) |
5 | 1, 4 | fndmexd 7844 | . . . 4 β’ (πΉ β π· β πΌ β V) |
6 | 2 | psrbag 21335 | . . . . 5 β’ (πΌ β V β (πΉ β π· β (πΉ:πΌβΆβ0 β§ (β‘πΉ β β) β Fin))) |
7 | 6 | biimpa 478 | . . . 4 β’ ((πΌ β V β§ πΉ β π·) β (πΉ:πΌβΆβ0 β§ (β‘πΉ β β) β Fin)) |
8 | 5, 7 | mpancom 687 | . . 3 β’ (πΉ β π· β (πΉ:πΌβΆβ0 β§ (β‘πΉ β β) β Fin)) |
9 | 8 | simprd 497 | . 2 β’ (πΉ β π· β (β‘πΉ β β) β Fin) |
10 | fcdmnn0fsuppg 12477 | . . 3 β’ ((πΉ β π· β§ πΉ:πΌβΆβ0) β (πΉ finSupp 0 β (β‘πΉ β β) β Fin)) | |
11 | 3, 10 | mpdan 686 | . 2 β’ (πΉ β π· β (πΉ finSupp 0 β (β‘πΉ β β) β Fin)) |
12 | 9, 11 | mpbird 257 | 1 β’ (πΉ β π· β πΉ finSupp 0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3406 Vcvv 3444 class class class wbr 5106 β‘ccnv 5633 β cima 5637 βΆwf 6493 (class class class)co 7358 βm cmap 8768 Fincfn 8886 finSupp cfsupp 9308 0cc0 11056 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-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 |
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-ov 7361 df-oprab 7362 df-mpo 7363 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-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fsupp 9309 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-nn 12159 df-n0 12419 |
This theorem is referenced by: psrbagaddcl 21346 psrbagev1 21501 mhpmulcl 21555 tdeglem1 25436 tdeglem3 25438 tdeglem4 25440 evlsbagval 40791 mhphflem 40813 mhphf 40814 |
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