![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5866 | . . . . 5 β’ (π = πΉ β β‘π = β‘πΉ) | |
2 | 1 | imaeq1d 6051 | . . . 4 β’ (π = πΉ β (β‘π β β) = (β‘πΉ β β)) |
3 | 2 | eleq1d 2812 | . . 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 3681 | . 2 β’ (πΉ β π· β (πΉ β (β0 βm πΌ) β§ (β‘πΉ β β) β Fin)) |
6 | nn0ex 12479 | . . . 4 β’ β0 β V | |
7 | elmapg 8832 | . . . 4 β’ ((β0 β V β§ πΌ β π) β (πΉ β (β0 βm πΌ) β πΉ:πΌβΆβ0)) | |
8 | 6, 7 | mpan 687 | . . 3 β’ (πΌ β π β (πΉ β (β0 βm πΌ) β πΉ:πΌβΆβ0)) |
9 | 8 | anbi1d 629 | . 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 1533 β wcel 2098 {crab 3426 Vcvv 3468 β‘ccnv 5668 β cima 5672 βΆwf 6532 (class class class)co 7404 βm cmap 8819 Fincfn 8938 βcn 12213 β0cn0 12473 |
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 7721 ax-cnex 11165 ax-1cn 11167 ax-addcl 11169 |
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-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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-map 8821 df-nn 12214 df-n0 12474 |
This theorem is referenced by: psrbagfOLD 21809 psrbagfsupp 21810 psrbagfsuppOLD 21811 snifpsrbag 21812 psrbaglecl 21816 psrbagleclOLD 21817 psrbagaddcl 21818 psrbagaddclOLD 21819 psrbagcon 21820 psrbagconOLD 21821 psrbaglefiOLD 21823 mplcoe5lem 21932 mplcoe5 21933 mplbas2 21935 psrbag0 21961 psrbagsn 21962 evlslem3 21981 mhpmulcl 22028 psrbagres 41653 evlselvlem 41696 evlselv 41697 |
Copyright terms: Public domain | W3C validator |