![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 2831 | . 2 ⊢ (𝐹 ∈ 𝐷 ↔ 𝐹 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) |
3 | elrabi 3690 | . . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝐹 ∈ (ℕ0 ↑m 𝐼)) | |
4 | elmapi 8888 | . . 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 1537 ∈ wcel 2106 {crab 3433 ◡ccnv 5688 “ cima 5692 ⟶wf 6559 (class class class)co 7431 ↑m cmap 8865 Fincfn 8984 ℕcn 12264 ℕ0cn0 12524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 |
This theorem is referenced by: psrbagfsupp 21957 psrbaglesupp 21960 psrbaglecl 21961 psrbagaddcl 21962 psrbagcon 21963 psrbaglefi 21964 psrbagconcl 21965 psrbagleadd1 21966 psrbagconf1o 21967 gsumbagdiaglem 21968 psrass1lem 21970 rhmpsrlem2 21979 psrlidm 22000 psrridm 22001 psrass1 22002 psrcom 22006 mplsubrglem 22042 mplmonmul 22072 psrbagev1 22119 evlslem3 22122 evlslem1 22124 mhpmulcl 22171 psdcl 22183 psdmplcl 22184 psdadd 22185 psdvsca 22186 psdmul 22188 psropprmul 22255 tdeglem1 26112 tdeglem3 26113 tdeglem4 26114 mdegmullem 26132 psrbagres 42533 evlsvvvallem 42548 evlsvvval 42550 selvvvval 42572 evlselvlem 42573 evlselv 42574 mhphflem 42583 mhphf 42584 |
Copyright terms: Public domain | W3C validator |