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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 5830 | . . . . 5 β’ (π = πΉ β β‘π = β‘πΉ) | |
2 | 1 | imaeq1d 6013 | . . . 4 β’ (π = πΉ β (β‘π β β) = (β‘πΉ β β)) |
3 | 2 | eleq1d 2819 | . . 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 3649 | . 2 β’ (πΉ β π· β (πΉ β (β0 βm πΌ) β§ (β‘πΉ β β) β Fin)) |
6 | nn0ex 12424 | . . . 4 β’ β0 β V | |
7 | elmapg 8781 | . . . 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 631 | . 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 397 = wceq 1542 β wcel 2107 {crab 3406 Vcvv 3444 β‘ccnv 5633 β cima 5637 βΆwf 6493 (class class class)co 7358 βm cmap 8768 Fincfn 8886 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-1cn 11114 ax-addcl 11116 |
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-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-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-map 8770 df-nn 12159 df-n0 12419 |
This theorem is referenced by: psrbagfOLD 21337 psrbagfsupp 21338 psrbagfsuppOLD 21339 snifpsrbag 21340 psrbaglecl 21344 psrbagleclOLD 21345 psrbagaddcl 21346 psrbagaddclOLD 21347 psrbagcon 21348 psrbagconOLD 21349 psrbaglefiOLD 21351 mplcoe5lem 21456 mplcoe5 21457 mplbas2 21459 psrbag0 21486 psrbagsn 21487 evlslem3 21506 mhpmulcl 21555 |
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