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Mirrors > Home > MPE Home > Th. List > psrbagf | Structured version Visualization version GIF version |
Description: A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 30-Jul-2024.) |
Ref | Expression |
---|---|
psrbag.d | β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} |
Ref | Expression |
---|---|
psrbagf | β’ (πΉ β π· β πΉ:πΌβΆβ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrbag.d | . . 3 β’ π· = {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} | |
2 | 1 | eleq2i 2826 | . 2 β’ (πΉ β π· β πΉ β {π β (β0 βm πΌ) β£ (β‘π β β) β Fin}) |
3 | elrabi 3678 | . . 3 β’ (πΉ β {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} β πΉ β (β0 βm πΌ)) | |
4 | elmapi 8843 | . . 3 β’ (πΉ β (β0 βm πΌ) β πΉ:πΌβΆβ0) | |
5 | 3, 4 | syl 17 | . 2 β’ (πΉ β {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} β πΉ:πΌβΆβ0) |
6 | 2, 5 | sylbi 216 | 1 β’ (πΉ β π· β πΉ:πΌβΆβ0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {crab 3433 β‘ccnv 5676 β cima 5680 βΆwf 6540 (class class class)co 7409 βm cmap 8820 Fincfn 8939 βcn 12212 β0cn0 12472 |
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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 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-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-map 8822 |
This theorem is referenced by: psrbagfsupp 21473 psrbaglesupp 21477 psrbaglecl 21479 psrbagaddcl 21481 psrbagcon 21483 psrbaglefi 21485 psrbagconcl 21487 psrbagconf1o 21489 gsumbagdiaglem 21494 psrass1lem 21496 psrmulcllem 21506 psrlidm 21523 psrridm 21524 psrass1 21525 psrcom 21529 mplsubrglem 21563 mplmonmul 21591 psrbagev1 21638 evlslem3 21643 evlslem1 21645 mhpmulcl 21692 psropprmul 21760 tdeglem1 25573 tdeglem3 25575 tdeglem4 25577 mdegmullem 25596 psrbagres 41115 rhmmpllem2 41122 rhmcomulmpl 41124 evlsvvvallem 41133 evlsvvval 41135 selvvvval 41157 evlselvlem 41158 evlselv 41159 mhphflem 41168 mhphf 41169 |
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