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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 | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
| Ref | Expression |
|---|---|
| psrbagf | ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | psrbag.d | . . 3 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐹 ∈ 𝐷 ↔ 𝐹 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) |
| 3 | elrabi 3687 | . . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝐹 ∈ (ℕ0 ↑m 𝐼)) | |
| 4 | elmapi 8889 | . . 3 ⊢ (𝐹 ∈ (ℕ0 ↑m 𝐼) → 𝐹:𝐼⟶ℕ0) | |
| 5 | 3, 4 | syl 17 | . 2 ⊢ (𝐹 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝐹:𝐼⟶ℕ0) |
| 6 | 2, 5 | sylbi 217 | 1 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 ◡ccnv 5684 “ cima 5688 ⟶wf 6557 (class class class)co 7431 ↑m cmap 8866 Fincfn 8985 ℕcn 12266 ℕ0cn0 12526 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 |
| This theorem is referenced by: psrbagfsupp 21939 psrbaglesupp 21942 psrbaglecl 21943 psrbagaddcl 21944 psrbagcon 21945 psrbaglefi 21946 psrbagconcl 21947 psrbagleadd1 21948 psrbagconf1o 21949 gsumbagdiaglem 21950 psrass1lem 21952 rhmpsrlem2 21961 psrlidm 21982 psrridm 21983 psrass1 21984 psrcom 21988 mplsubrglem 22024 mplmonmul 22054 psrbagev1 22101 evlslem3 22104 evlslem1 22106 mhpmulcl 22153 psdcl 22165 psdmplcl 22166 psdadd 22167 psdvsca 22168 psdmul 22170 psdmvr 22173 psropprmul 22239 tdeglem1 26097 tdeglem3 26098 tdeglem4 26099 mdegmullem 26117 psrbagres 42556 evlsvvvallem 42571 evlsvvval 42573 selvvvval 42595 evlselvlem 42596 evlselv 42597 mhphflem 42606 mhphf 42607 |
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