psrbag0 - Metamath Proof Explorer

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Theorem psrbag0 20733

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 11900	. . . 4 ⊢ 0 ∈ ℕ₀
2	1	fconst6 6543	. . 3 ⊢ (𝐼 × {0}):𝐼⟶ℕ₀
3		c0ex 10624	. . . . . 6 ⊢ 0 ∈ V
4	3	fconst 6539	. . . . 5 ⊢ (𝐼 × {0}):𝐼⟶{0}
5		incom 4128	. . . . . 6 ⊢ ({0} ∩ ℕ) = (ℕ ∩ {0})
6		0nnn 11661	. . . . . . 7 ⊢ ¬ 0 ∈ ℕ
7		disjsn 4607	. . . . . . 7 ⊢ ((ℕ ∩ {0}) = ∅ ↔ ¬ 0 ∈ ℕ)
8	6, 7	mpbir 234	. . . . . 6 ⊢ (ℕ ∩ {0}) = ∅
9	5, 8	eqtri 2821	. . . . 5 ⊢ ({0} ∩ ℕ) = ∅
10		fimacnvdisj 6531	. . . . 5 ⊢ (((𝐼 × {0}):𝐼⟶{0} ∧ ({0} ∩ ℕ) = ∅) → (^◡(𝐼 × {0}) “ ℕ) = ∅)
11	4, 9, 10	mp2an 691	. . . 4 ⊢ (^◡(𝐼 × {0}) “ ℕ) = ∅
12		0fin 8730	. . . 4 ⊢ ∅ ∈ Fin
13	11, 12	eqeltri 2886	. . 3 ⊢ (^◡(𝐼 × {0}) “ ℕ) ∈ Fin
14	2, 13	pm3.2i 474	. 2 ⊢ ((𝐼 × {0}):𝐼⟶ℕ₀ ∧ (^◡(𝐼 × {0}) “ ℕ) ∈ Fin)
15		psrbag0.d	. . 3 ⊢ 𝐷 = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
16	15	psrbag 20602	. 2 ⊢ (𝐼 ∈ 𝑉 → ((𝐼 × {0}) ∈ 𝐷 ↔ ((𝐼 × {0}):𝐼⟶ℕ₀ ∧ (^◡(𝐼 × {0}) “ ℕ) ∈ Fin)))
17	14, 16	mpbiri 261	1 ⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) ∈ 𝐷)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {crab 3110 ∩ cin 3880 ∅c0 4243 {csn 4525 × cxp 5517 ^◡ccnv 5518 “ cima 5522 ⟶wf 6320 (class class class)co 7135 ↑_m cmap 8389 Fincfn 8492 0cc0 10526 ℕcn 11625 ℕ₀cn0 11885

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-nn 11626 df-n0 11886

This theorem is referenced by: mplascl 20735 subrgasclcl 20738 evlslem1 20754 tdeglem4 24661 mdegle0 24678

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