Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fczpsrbag | Structured version Visualization version GIF version |
Description: The constant function equal to zero is a finite bag. (Contributed by AV, 8-Jul-2019.) |
Ref | Expression |
---|---|
psrbag.d | ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} |
Ref | Expression |
---|---|
fczpsrbag | ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ 𝐼 ↦ 0) ∈ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifid 4513 | . . . . 5 ⊢ if(𝑥 = 𝑛, 0, 0) = 0 | |
2 | 1 | eqcomi 2745 | . . . 4 ⊢ 0 = if(𝑥 = 𝑛, 0, 0) |
3 | 2 | a1i 11 | . . 3 ⊢ (𝐼 ∈ 𝑉 → 0 = if(𝑥 = 𝑛, 0, 0)) |
4 | 3 | mpteq2dv 5194 | . 2 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ 𝐼 ↦ 0) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑛, 0, 0))) |
5 | 0nn0 12349 | . . 3 ⊢ 0 ∈ ℕ0 | |
6 | psrbag.d | . . . 4 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
7 | 6 | snifpsrbag 21231 | . . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 0 ∈ ℕ0) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑛, 0, 0)) ∈ 𝐷) |
8 | 5, 7 | mpan2 688 | . 2 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑛, 0, 0)) ∈ 𝐷) |
9 | 4, 8 | eqeltrd 2837 | 1 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ 𝐼 ↦ 0) ∈ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 {crab 3403 ifcif 4473 ↦ cmpt 5175 ◡ccnv 5619 “ cima 5623 (class class class)co 7337 ↑m cmap 8686 Fincfn 8804 0cc0 10972 ℕcn 12074 ℕ0cn0 12334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-2nd 7900 df-supp 8048 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-fsupp 9227 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-nn 12075 df-n0 12335 |
This theorem is referenced by: psrbas 21253 psrlidm 21278 psrridm 21279 |
Copyright terms: Public domain | W3C validator |