psrbag0 - Metamath Proof Explorer

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Theorem psrbag0 21315

Description: The empty bag is a bag. (Contributed by Stefan O'Rear, 9-Mar-2015.)

**Hypothesis**
Ref	Expression
psrbag0.d	⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}

**Assertion**
Ref	Expression
psrbag0	⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) ∈ 𝐷)

Distinct variable group: 𝑓,𝐼 Allowed substitution hints: 𝐷(𝑓) 𝑉(𝑓)

**Proof of Theorem psrbag0**
Step	Hyp	Ref	Expression
1		0nn0 12294	. . . 4 ⊢ 0 ∈ ℕ₀
2	1	fconst6 6694	. . 3 ⊢ (𝐼 × {0}):𝐼⟶ℕ₀
3		c0ex 11015	. . . . . 6 ⊢ 0 ∈ V
4	3	fconst 6690	. . . . 5 ⊢ (𝐼 × {0}):𝐼⟶{0}
5		incom 4141	. . . . . 6 ⊢ ({0} ∩ ℕ) = (ℕ ∩ {0})
6		0nnn 12055	. . . . . . 7 ⊢ ¬ 0 ∈ ℕ
7		disjsn 4651	. . . . . . 7 ⊢ ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
8	6, 7	mpbir 230	. . . . . 6 ⊢ (ℕ ∩ {0}) = ∅
9	5, 8	eqtri 2764	. . . . 5 ⊢ ({0} ∩ ℕ) = ∅
10		fimacnvdisj 6682	. . . . 5 ⊢ (((𝐼 × {0}):𝐼⟶{0} ∧ ({0} ∩ ℕ) = ∅) → (^◡(𝐼 × {0}) “ ℕ) = ∅)
11	4, 9, 10	mp2an 690	. . . 4 ⊢ (^◡(𝐼 × {0}) “ ℕ) = ∅
12		0fin 8992	. . . 4 ⊢ ∅ ∈ Fin
13	11, 12	eqeltri 2833	. . 3 ⊢ (^◡(𝐼 × {0}) “ ℕ) ∈ Fin
14	2, 13	pm3.2i 472	. 2 ⊢ ((𝐼 × {0}):𝐼⟶ℕ₀ ∧ (^◡(𝐼 × {0}) “ ℕ) ∈ Fin)
15		psrbag0.d	. . 3 ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
16	15	psrbag 21165	. 2 ⊢ (𝐼 ∈ 𝑉 → ((𝐼 × {0}) ∈ 𝐷 ↔ ((𝐼 × {0}):𝐼⟶ℕ₀ ∧ (^◡(𝐼 × {0}) “ ℕ) ∈ Fin)))
17	14, 16	mpbiri 258	1 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) ∈ 𝐷)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1539 ∈ wcel 2104 {crab 3284 ∩ cin 3891 ∅c0 4262 {csn 4565 × cxp 5598 ^◡ccnv 5599 “ cima 5603 ⟶wf 6454 (class class class)co 7307 ↑_m cmap 8646 Fincfn 8764 0cc0 10917 ℕcn 12019 ℕ₀cn0 12279

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 ax-cnex 10973 ax-resscn 10974 ax-1cn 10975 ax-icn 10976 ax-addcl 10977 ax-addrcl 10978 ax-mulcl 10979 ax-mulrcl 10980 ax-mulcom 10981 ax-addass 10982 ax-mulass 10983 ax-distr 10984 ax-i2m1 10985 ax-1ne0 10986 ax-1rid 10987 ax-rnegex 10988 ax-rrecex 10989 ax-cnre 10990 ax-pre-lttri 10991 ax-pre-lttrn 10992 ax-pre-ltadd 10993

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-pred 6217 df-ord 6284 df-on 6285 df-lim 6286 df-suc 6287 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-fv 6466 df-ov 7310 df-oprab 7311 df-mpo 7312 df-om 7745 df-2nd 7864 df-frecs 8128 df-wrecs 8159 df-recs 8233 df-rdg 8272 df-er 8529 df-map 8648 df-en 8765 df-dom 8766 df-sdom 8767 df-fin 8768 df-pnf 11057 df-mnf 11058 df-xr 11059 df-ltxr 11060 df-le 11061 df-nn 12020 df-n0 12280

This theorem is referenced by: mplascl 21317 subrgasclcl 21320 evlslem1 21337 tdeglem4 25269 tdeglem4OLD 25270 mdegle0 25287

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