psrbag0 - Metamath Proof Explorer

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Theorem psrbag0 22050

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 12443	. . . 4 ⊢ 0 ∈ ℕ₀
2	1	fconst6 6724	. . 3 ⊢ (𝐼 × {0}):𝐼⟶ℕ₀
3		c0ex 11129	. . . . . 6 ⊢ 0 ∈ V
4	3	fconst 6720	. . . . 5 ⊢ (𝐼 × {0}):𝐼⟶{0}
5		incom 4150	. . . . . 6 ⊢ ({0} ∩ ℕ) = (ℕ ∩ {0})
6		0nnn 12204	. . . . . . 7 ⊢ ¬ 0 ∈ ℕ
7		disjsn 4656	. . . . . . 7 ⊢ ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
8	6, 7	mpbir 231	. . . . . 6 ⊢ (ℕ ∩ {0}) = ∅
9	5, 8	eqtri 2760	. . . . 5 ⊢ ({0} ∩ ℕ) = ∅
10		fimacnvdisj 6712	. . . . 5 ⊢ (((𝐼 × {0}):𝐼⟶{0} ∧ ({0} ∩ ℕ) = ∅) → (^◡(𝐼 × {0}) “ ℕ) = ∅)
11	4, 9, 10	mp2an 693	. . . 4 ⊢ (^◡(𝐼 × {0}) “ ℕ) = ∅
12		0fi 8982	. . . 4 ⊢ ∅ ∈ Fin
13	11, 12	eqeltri 2833	. . 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 21907	. 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 1542 ∈ wcel 2114 {crab 3390 ∩ cin 3889 ∅c0 4274 {csn 4568 × cxp 5622 ^◡ccnv 5623 “ cima 5627 ⟶wf 6488 (class class class)co 7360 ↑_m cmap 8766 Fincfn 8886 0cc0 11029 ℕcn 12165 ℕ₀cn0 12428

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-nn 12166 df-n0 12429

This theorem is referenced by: mplascl 22052 subrgasclcl 22055 evlslem1 22070 tdeglem4 26035 mdegle0 26052

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