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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 | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
Ref | Expression |
---|---|
psrbag | ⊢ (𝐼 ∈ 𝑉 → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveq 5864 | . . . . 5 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹) | |
2 | 1 | imaeq1d 6049 | . . . 4 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ ℕ) = (◡𝐹 “ ℕ)) |
3 | 2 | eleq1d 2810 | . . 3 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ ℕ) ∈ Fin ↔ (◡𝐹 “ ℕ) ∈ Fin)) |
4 | psrbag.d | . . 3 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
5 | 3, 4 | elrab2 3679 | . 2 ⊢ (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ (ℕ0 ↑m 𝐼) ∧ (◡𝐹 “ ℕ) ∈ Fin)) |
6 | nn0ex 12476 | . . . 4 ⊢ ℕ0 ∈ V | |
7 | elmapg 8830 | . . . 4 ⊢ ((ℕ0 ∈ V ∧ 𝐼 ∈ 𝑉) → (𝐹 ∈ (ℕ0 ↑m 𝐼) ↔ 𝐹:𝐼⟶ℕ0)) | |
8 | 6, 7 | mpan 687 | . . 3 ⊢ (𝐼 ∈ 𝑉 → (𝐹 ∈ (ℕ0 ↑m 𝐼) ↔ 𝐹:𝐼⟶ℕ0)) |
9 | 8 | anbi1d 629 | . 2 ⊢ (𝐼 ∈ 𝑉 → ((𝐹 ∈ (ℕ0 ↑m 𝐼) ∧ (◡𝐹 “ ℕ) ∈ Fin) ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin))) |
10 | 5, 9 | bitrid 283 | 1 ⊢ (𝐼 ∈ 𝑉 → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3424 Vcvv 3466 ◡ccnv 5666 “ cima 5670 ⟶wf 6530 (class class class)co 7402 ↑m cmap 8817 Fincfn 8936 ℕcn 12210 ℕ0cn0 12470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-1cn 11165 ax-addcl 11167 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-map 8819 df-nn 12211 df-n0 12471 |
This theorem is referenced by: psrbagfOLD 21783 psrbagfsupp 21784 psrbagfsuppOLD 21785 snifpsrbag 21786 psrbaglecl 21790 psrbagleclOLD 21791 psrbagaddcl 21792 psrbagaddclOLD 21793 psrbagcon 21794 psrbagconOLD 21795 psrbaglefiOLD 21797 mplcoe5lem 21906 mplcoe5 21907 mplbas2 21909 psrbag0 21935 psrbagsn 21936 evlslem3 21955 mhpmulcl 22002 psrbagres 41608 evlselvlem 41651 evlselv 41652 |
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