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Mirrors > Home > MPE Home > Th. List > psrbag | Structured version Visualization version GIF version |
Description: Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
psrbag.d | β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} |
Ref | Expression |
---|---|
psrbag | β’ (πΌ β π β (πΉ β π· β (πΉ:πΌβΆβ0 β§ (β‘πΉ β β) β Fin))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveq 5876 | . . . . 5 β’ (π = πΉ β β‘π = β‘πΉ) | |
2 | 1 | imaeq1d 6062 | . . . 4 β’ (π = πΉ β (β‘π β β) = (β‘πΉ β β)) |
3 | 2 | eleq1d 2814 | . . 3 β’ (π = πΉ β ((β‘π β β) β Fin β (β‘πΉ β β) β Fin)) |
4 | psrbag.d | . . 3 β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} | |
5 | 3, 4 | elrab2 3685 | . 2 β’ (πΉ β π· β (πΉ β (β0 βm πΌ) β§ (β‘πΉ β β) β Fin)) |
6 | nn0ex 12509 | . . . 4 β’ β0 β V | |
7 | elmapg 8858 | . . . 4 β’ ((β0 β V β§ πΌ β π) β (πΉ β (β0 βm πΌ) β πΉ:πΌβΆβ0)) | |
8 | 6, 7 | mpan 689 | . . 3 β’ (πΌ β π β (πΉ β (β0 βm πΌ) β πΉ:πΌβΆβ0)) |
9 | 8 | anbi1d 630 | . 2 β’ (πΌ β π β ((πΉ β (β0 βm πΌ) β§ (β‘πΉ β β) β Fin) β (πΉ:πΌβΆβ0 β§ (β‘πΉ β β) β Fin))) |
10 | 5, 9 | bitrid 283 | 1 β’ (πΌ β π β (πΉ β π· β (πΉ:πΌβΆβ0 β§ (β‘πΉ β β) β Fin))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 {crab 3429 Vcvv 3471 β‘ccnv 5677 β cima 5681 βΆwf 6544 (class class class)co 7420 βm cmap 8845 Fincfn 8964 βcn 12243 β0cn0 12503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-1cn 11197 ax-addcl 11199 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-map 8847 df-nn 12244 df-n0 12504 |
This theorem is referenced by: psrbagfOLD 21852 psrbagfsupp 21853 psrbagfsuppOLD 21854 snifpsrbag 21855 psrbaglecl 21859 psrbagleclOLD 21860 psrbagaddcl 21861 psrbagaddclOLD 21862 psrbagcon 21863 psrbagconOLD 21864 psrbaglefiOLD 21866 mplcoe5lem 21977 mplcoe5 21978 mplbas2 21980 psrbag0 22006 psrbagsn 22007 evlslem3 22026 mhpmulcl 22073 psrbagres 41776 evlselvlem 41819 evlselv 41820 |
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