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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 3640 | . . 3 β’ (πΉ β {π β (β0 βm πΌ) β£ (β‘π β β) β Fin} β πΉ β (β0 βm πΌ)) | |
4 | elmapi 8790 | . . 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 3406 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 |
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 3062 df-rex 3071 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-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-id 5532 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-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-map 8770 |
This theorem is referenced by: psrbagfsupp 21338 psrbaglesupp 21342 psrbaglecl 21344 psrbagaddcl 21346 psrbagcon 21348 psrbaglefi 21350 psrbagconcl 21352 psrbagconf1o 21354 gsumbagdiaglem 21359 psrass1lem 21361 psrmulcllem 21371 psrlidm 21388 psrridm 21389 psrass1 21390 psrcom 21394 mplsubrglem 21426 mplmonmul 21453 psrbagev1 21501 evlslem3 21506 evlslem1 21508 mhpmulcl 21555 psropprmul 21625 tdeglem1 25436 tdeglem3 25438 tdeglem4 25440 mdegmullem 25459 rhmmpllem2 40781 rhmcomulmpl 40783 evlsbagval 40791 mhphflem 40813 |
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