psrbag0 - Metamath Proof Explorer

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Theorem psrbag0 21951

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 12387	. . . 4 ⊢ 0 ∈ ℕ₀
2	1	fconst6 6708	. . 3 ⊢ (𝐼 × {0}):𝐼⟶ℕ₀
3		c0ex 11097	. . . . . 6 ⊢ 0 ∈ V
4	3	fconst 6704	. . . . 5 ⊢ (𝐼 × {0}):𝐼⟶{0}
5		incom 4156	. . . . . 6 ⊢ ({0} ∩ ℕ) = (ℕ ∩ {0})
6		0nnn 12152	. . . . . . 7 ⊢ ¬ 0 ∈ ℕ
7		disjsn 4661	. . . . . . 7 ⊢ ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
8	6, 7	mpbir 231	. . . . . 6 ⊢ (ℕ ∩ {0}) = ∅
9	5, 8	eqtri 2752	. . . . 5 ⊢ ({0} ∩ ℕ) = ∅
10		fimacnvdisj 6696	. . . . 5 ⊢ (((𝐼 × {0}):𝐼⟶{0} ∧ ({0} ∩ ℕ) = ∅) → (^◡(𝐼 × {0}) “ ℕ) = ∅)
11	4, 9, 10	mp2an 692	. . . 4 ⊢ (^◡(𝐼 × {0}) “ ℕ) = ∅
12		0fi 8958	. . . 4 ⊢ ∅ ∈ Fin
13	11, 12	eqeltri 2824	. . 3 ⊢ (^◡(𝐼 × {0}) “ ℕ) ∈ Fin
14	2, 13	pm3.2i 470	. 2 ⊢ ((𝐼 × {0}):𝐼⟶ℕ₀ ∧ (^◡(𝐼 × {0}) “ ℕ) ∈ Fin)
15		psrbag0.d	. . 3 ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
16	15	psrbag 21808	. 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 395 = wceq 1540 ∈ wcel 2109 {crab 3392 ∩ cin 3898 ∅c0 4280 {csn 4573 × cxp 5611 ^◡ccnv 5612 “ cima 5616 ⟶wf 6472 (class class class)co 7340 ↑_m cmap 8744 Fincfn 8863 0cc0 10997 ℕcn 12116 ℕ₀cn0 12372

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5367 ax-un 7662 ax-cnex 11053 ax-resscn 11054 ax-1cn 11055 ax-icn 11056 ax-addcl 11057 ax-addrcl 11058 ax-mulcl 11059 ax-mulrcl 11060 ax-mulcom 11061 ax-addass 11062 ax-mulass 11063 ax-distr 11064 ax-i2m1 11065 ax-1ne0 11066 ax-1rid 11067 ax-rnegex 11068 ax-rrecex 11069 ax-cnre 11070 ax-pre-lttri 11071 ax-pre-lttrn 11072 ax-pre-ltadd 11073

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3393 df-v 3435 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4940 df-br 5089 df-opab 5151 df-mpt 5170 df-tr 5196 df-id 5508 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5566 df-we 5568 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-rn 5624 df-res 5625 df-ima 5626 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-ov 7343 df-oprab 7344 df-mpo 7345 df-om 7791 df-2nd 7916 df-frecs 8205 df-wrecs 8236 df-recs 8285 df-rdg 8323 df-er 8616 df-map 8746 df-en 8864 df-dom 8865 df-sdom 8866 df-fin 8867 df-pnf 11139 df-mnf 11140 df-xr 11141 df-ltxr 11142 df-le 11143 df-nn 12117 df-n0 12373

This theorem is referenced by: mplascl 21953 subrgasclcl 21956 evlslem1 21971 tdeglem4 25946 mdegle0 25963

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