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Mirrors > Home > MPE Home > Th. List > psrbaglecl | Structured version Visualization version GIF version |
Description: The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.) |
Ref | Expression |
---|---|
psrbag.d | β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} |
Ref | Expression |
---|---|
psrbaglecl | β’ ((πΉ β π· β§ πΊ:πΌβΆβ0 β§ πΊ βr β€ πΉ) β πΊ β π·) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1134 | . 2 β’ ((πΉ β π· β§ πΊ:πΌβΆβ0 β§ πΊ βr β€ πΉ) β πΊ:πΌβΆβ0) | |
2 | simp1 1133 | . . . . 5 β’ ((πΉ β π· β§ πΊ:πΌβΆβ0 β§ πΊ βr β€ πΉ) β πΉ β π·) | |
3 | id 22 | . . . . . . . 8 β’ (πΉ β π· β πΉ β π·) | |
4 | psrbag.d | . . . . . . . . . 10 β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} | |
5 | 4 | psrbagf 21812 | . . . . . . . . 9 β’ (πΉ β π· β πΉ:πΌβΆβ0) |
6 | 5 | ffnd 6712 | . . . . . . . 8 β’ (πΉ β π· β πΉ Fn πΌ) |
7 | 3, 6 | fndmexd 7894 | . . . . . . 7 β’ (πΉ β π· β πΌ β V) |
8 | 7 | 3ad2ant1 1130 | . . . . . 6 β’ ((πΉ β π· β§ πΊ:πΌβΆβ0 β§ πΊ βr β€ πΉ) β πΌ β V) |
9 | 4 | psrbag 21811 | . . . . . 6 β’ (πΌ β V β (πΉ β π· β (πΉ:πΌβΆβ0 β§ (β‘πΉ β β) β Fin))) |
10 | 8, 9 | syl 17 | . . . . 5 β’ ((πΉ β π· β§ πΊ:πΌβΆβ0 β§ πΊ βr β€ πΉ) β (πΉ β π· β (πΉ:πΌβΆβ0 β§ (β‘πΉ β β) β Fin))) |
11 | 2, 10 | mpbid 231 | . . . 4 β’ ((πΉ β π· β§ πΊ:πΌβΆβ0 β§ πΊ βr β€ πΉ) β (πΉ:πΌβΆβ0 β§ (β‘πΉ β β) β Fin)) |
12 | 11 | simprd 495 | . . 3 β’ ((πΉ β π· β§ πΊ:πΌβΆβ0 β§ πΊ βr β€ πΉ) β (β‘πΉ β β) β Fin) |
13 | 4 | psrbaglesupp 21818 | . . 3 β’ ((πΉ β π· β§ πΊ:πΌβΆβ0 β§ πΊ βr β€ πΉ) β (β‘πΊ β β) β (β‘πΉ β β)) |
14 | 12, 13 | ssfid 9269 | . 2 β’ ((πΉ β π· β§ πΊ:πΌβΆβ0 β§ πΊ βr β€ πΉ) β (β‘πΊ β β) β Fin) |
15 | 4 | psrbag 21811 | . . 3 β’ (πΌ β V β (πΊ β π· β (πΊ:πΌβΆβ0 β§ (β‘πΊ β β) β Fin))) |
16 | 8, 15 | syl 17 | . 2 β’ ((πΉ β π· β§ πΊ:πΌβΆβ0 β§ πΊ βr β€ πΉ) β (πΊ β π· β (πΊ:πΌβΆβ0 β§ (β‘πΊ β β) β Fin))) |
17 | 1, 14, 16 | mpbir2and 710 | 1 β’ ((πΉ β π· β§ πΊ:πΌβΆβ0 β§ πΊ βr β€ πΉ) β πΊ β π·) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 {crab 3426 Vcvv 3468 class class class wbr 5141 β‘ccnv 5668 β cima 5672 βΆwf 6533 (class class class)co 7405 βr cofr 7666 βm cmap 8822 Fincfn 8941 β€ cle 11253 βcn 12216 β0cn0 12476 |
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 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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-nel 3041 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-iun 4992 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-pred 6294 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-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 |
This theorem is referenced by: psrbaglefi 21826 |
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