psrbag0 - Metamath Proof Explorer

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Theorem psrbag0 22040

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 12452	. . . 4 ⊢ 0 ∈ ℕ₀
2	1	fconst6 6730	. . 3 ⊢ (𝐼 × {0}):𝐼⟶ℕ₀
3		c0ex 11138	. . . . . 6 ⊢ 0 ∈ V
4	3	fconst 6726	. . . . 5 ⊢ (𝐼 × {0}):𝐼⟶{0}
5		incom 4149	. . . . . 6 ⊢ ({0} ∩ ℕ) = (ℕ ∩ {0})
6		0nnn 12213	. . . . . . 7 ⊢ ¬ 0 ∈ ℕ
7		disjsn 4655	. . . . . . 7 ⊢ ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
8	6, 7	mpbir 231	. . . . . 6 ⊢ (ℕ ∩ {0}) = ∅
9	5, 8	eqtri 2759	. . . . 5 ⊢ ({0} ∩ ℕ) = ∅
10		fimacnvdisj 6718	. . . . 5 ⊢ (((𝐼 × {0}):𝐼⟶{0} ∧ ({0} ∩ ℕ) = ∅) → (^◡(𝐼 × {0}) “ ℕ) = ∅)
11	4, 9, 10	mp2an 693	. . . 4 ⊢ (^◡(𝐼 × {0}) “ ℕ) = ∅
12		0fi 8989	. . . 4 ⊢ ∅ ∈ Fin
13	11, 12	eqeltri 2832	. . 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 21897	. 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 3389 ∩ cin 3888 ∅c0 4273 {csn 4567 × cxp 5629 ^◡ccnv 5630 “ cima 5634 ⟶wf 6494 (class class class)co 7367 ↑_m cmap 8773 Fincfn 8893 0cc0 11038 ℕcn 12174 ℕ₀cn0 12437

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 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-nn 12175 df-n0 12438

This theorem is referenced by: mplascl 22042 subrgasclcl 22045 evlslem1 22060 tdeglem4 26025 mdegle0 26042

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