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Mirrors > Home > MPE Home > Th. List > snifpsrbag | Structured version Visualization version GIF version |
Description: A bag containing one element is a finite bag. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 8-Jul-2019.) |
Ref | Expression |
---|---|
psrbag.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
Ref | Expression |
---|---|
snifpsrbag | ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
2 | 0nn0 12485 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
3 | 2 | a1i 11 | . . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 0 ∈ ℕ0) |
4 | 1, 3 | ifcld 4567 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → if(𝑦 = 𝑋, 𝑁, 0) ∈ ℕ0) |
5 | 4 | adantr 480 | . . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ 𝑦 ∈ 𝐼) → if(𝑦 = 𝑋, 𝑁, 0) ∈ ℕ0) |
6 | 5 | fmpttd 7107 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)):𝐼⟶ℕ0) |
7 | id 22 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ 𝑉) | |
8 | c0ex 11206 | . . . . . 6 ⊢ 0 ∈ V | |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝐼 ∈ 𝑉 → 0 ∈ V) |
10 | eqid 2724 | . . . . 5 ⊢ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) | |
11 | 7, 9, 10 | sniffsupp 9392 | . . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) finSupp 0) |
12 | 11 | adantr 480 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) finSupp 0) |
13 | fcdmnn0fsupp 12527 | . . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)):𝐼⟶ℕ0) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) finSupp 0 ↔ (◡(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) “ ℕ) ∈ Fin)) | |
14 | 13 | adantlr 712 | . . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)):𝐼⟶ℕ0) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) finSupp 0 ↔ (◡(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) “ ℕ) ∈ Fin)) |
15 | 14 | bicomd 222 | . . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)):𝐼⟶ℕ0) → ((◡(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) “ ℕ) ∈ Fin ↔ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) finSupp 0)) |
16 | 6, 15 | mpdan 684 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((◡(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) “ ℕ) ∈ Fin ↔ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) finSupp 0)) |
17 | 12, 16 | mpbird 257 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (◡(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) “ ℕ) ∈ Fin) |
18 | psrbag.d | . . . 4 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
19 | 18 | psrbag 21781 | . . 3 ⊢ (𝐼 ∈ 𝑉 → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) ∈ 𝐷 ↔ ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)):𝐼⟶ℕ0 ∧ (◡(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) “ ℕ) ∈ Fin))) |
20 | 19 | adantr 480 | . 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) ∈ 𝐷 ↔ ((𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)):𝐼⟶ℕ0 ∧ (◡(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) “ ℕ) ∈ Fin))) |
21 | 6, 17, 20 | mpbir2and 710 | 1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 𝑁, 0)) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3424 Vcvv 3466 ifcif 4521 class class class wbr 5139 ↦ cmpt 5222 ◡ccnv 5666 “ cima 5670 ⟶wf 6530 (class class class)co 7402 ↑m cmap 8817 Fincfn 8936 finSupp cfsupp 9358 0cc0 11107 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-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 |
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-nel 3039 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-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-nn 12211 df-n0 12471 |
This theorem is referenced by: fczpsrbag 21787 mvrid 21855 mvrf1 21857 mplcoe3 21905 mplcoe5 21907 psdcl 22014 |
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