Theorem psrbaglecl 14711

Description: The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)

Hypothesis

Ref	Expression
psrbag.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}

Assertion

Ref	Expression
psrbaglecl	⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺 ∈ 𝐷)

Distinct variable groups:   𝑓,𝐹   𝑓,𝐼   𝑓,𝐺

Allowed substitution hint:   𝐷(𝑓)

Proof of Theorem psrbaglecl

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1024	. 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺:𝐼⟶ℕ0)
2		simp1 1023	. . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐹 ∈ 𝐷)
3		id 19	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐷 → 𝐹 ∈ 𝐷)
4		psrbag.d	. . . . . . . . . 10 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
5	4	psrbagf 14706	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
6	5	ffnd 5483	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼)
7	3, 6	fndmexd 5526	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → 𝐼 ∈ V)
8	7	3ad2ant1 1044	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐼 ∈ V)
9	4	psrbag 14705	. . . . . 6 ⊢ (𝐼 ∈ V → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin)))
10	8, 9	syl 14	. . . . 5 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin)))
11	2, 10	mpbid 147	. . . 4 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin))
12	11	simprd 114	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (◡𝐹 “ ℕ) ∈ Fin)
13	4	psrbaglesupp 14709	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))
14	1	adantr 276	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (◡𝐹 “ ℕ)) → 𝐺:𝐼⟶ℕ0)
15	5	3ad2ant1 1044	. . . . . . . . . 10 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐹:𝐼⟶ℕ0)
16		ffn 5482	. . . . . . . . . 10 ⊢ (𝐹:𝐼⟶ℕ0 → 𝐹 Fn 𝐼)
17		elpreima 5767	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐼 → (𝑥 ∈ (◡𝐹 “ ℕ) ↔ (𝑥 ∈ 𝐼 ∧ (𝐹‘𝑥) ∈ ℕ)))
18	15, 16, 17	3syl 17	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝑥 ∈ (◡𝐹 “ ℕ) ↔ (𝑥 ∈ 𝐼 ∧ (𝐹‘𝑥) ∈ ℕ)))
19	18	simprbda 383	. . . . . . . 8 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (◡𝐹 “ ℕ)) → 𝑥 ∈ 𝐼)
20	14, 19	ffvelcdmd 5784	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (◡𝐹 “ ℕ)) → (𝐺‘𝑥) ∈ ℕ0)
21	20	nn0zd 9603	. . . . . 6 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (◡𝐹 “ ℕ)) → (𝐺‘𝑥) ∈ ℤ)
22		elnndc 9849	. . . . . 6 ⊢ ((𝐺‘𝑥) ∈ ℤ → DECID (𝐺‘𝑥) ∈ ℕ)
23	21, 22	syl 14	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (◡𝐹 “ ℕ)) → DECID (𝐺‘𝑥) ∈ ℕ)
24		ffn 5482	. . . . . . . . 9 ⊢ (𝐺:𝐼⟶ℕ0 → 𝐺 Fn 𝐼)
25		elpreima 5767	. . . . . . . . 9 ⊢ (𝐺 Fn 𝐼 → (𝑥 ∈ (◡𝐺 “ ℕ) ↔ (𝑥 ∈ 𝐼 ∧ (𝐺‘𝑥) ∈ ℕ)))
26	1, 24, 25	3syl 17	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝑥 ∈ (◡𝐺 “ ℕ) ↔ (𝑥 ∈ 𝐼 ∧ (𝐺‘𝑥) ∈ ℕ)))
27	26	adantr 276	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (◡𝐹 “ ℕ)) → (𝑥 ∈ (◡𝐺 “ ℕ) ↔ (𝑥 ∈ 𝐼 ∧ (𝐺‘𝑥) ∈ ℕ)))
28	19, 27	mpbirand 441	. . . . . 6 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (◡𝐹 “ ℕ)) → (𝑥 ∈ (◡𝐺 “ ℕ) ↔ (𝐺‘𝑥) ∈ ℕ))
29	28	dcbid 845	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (◡𝐹 “ ℕ)) → (DECID 𝑥 ∈ (◡𝐺 “ ℕ) ↔ DECID (𝐺‘𝑥) ∈ ℕ))
30	23, 29	mpbird 167	. . . 4 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (◡𝐹 “ ℕ)) → DECID 𝑥 ∈ (◡𝐺 “ ℕ))
31	30	ralrimiva 2605	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ∀𝑥 ∈ (◡𝐹 “ ℕ)DECID 𝑥 ∈ (◡𝐺 “ ℕ))
32		ssfidc 7133	. . 3 ⊢ (((◡𝐹 “ ℕ) ∈ Fin ∧ (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ) ∧ ∀𝑥 ∈ (◡𝐹 “ ℕ)DECID 𝑥 ∈ (◡𝐺 “ ℕ)) → (◡𝐺 “ ℕ) ∈ Fin)
33	12, 13, 31, 32	syl3anc 1273	. 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (◡𝐺 “ ℕ) ∈ Fin)
34	4	psrbag 14705	. . 3 ⊢ (𝐼 ∈ V → (𝐺 ∈ 𝐷 ↔ (𝐺:𝐼⟶ℕ0 ∧ (◡𝐺 “ ℕ) ∈ Fin)))
35	8, 34	syl 14	. 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (𝐺 ∈ 𝐷 ↔ (𝐺:𝐼⟶ℕ0 ∧ (◡𝐺 “ ℕ) ∈ Fin)))
36	1, 33, 35	mpbir2and 952	1 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺 ∈ 𝐷)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 841   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510  {crab 2514  Vcvv 2802   ⊆ wss 3200   class class class wbr 4088  ^◡ccnv 4724   “ cima 4728   Fn wfn 5321  ⟶wf 5322  ‘cfv 5326  (class class class)co 6021   ∘_𝑟 cofr 6237   ↑_𝑚 cmap 6820  Fincfn 6912   ≤ cle 8218  ℕcn 9146  ℕ₀cn0 9405  ℤcz 9482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8126  ax-resscn 8127  ax-1cn 8128  ax-1re 8129  ax-icn 8130  ax-addcl 8131  ax-addrcl 8132  ax-mulcl 8133  ax-addcom 8135  ax-addass 8137  ax-distr 8139  ax-i2m1 8140  ax-0lt1 8141  ax-0id 8143  ax-rnegex 8144  ax-cnre 8146  ax-pre-ltirr 8147  ax-pre-ltwlin 8148  ax-pre-lttrn 8149  ax-pre-ltadd 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5974  df-ov 6024  df-oprab 6025  df-mpo 6026  df-ofr 6239  df-1o 6585  df-er 6705  df-map 6822  df-en 6913  df-fin 6915  df-pnf 8219  df-mnf 8220  df-xr 8221  df-ltxr 8222  df-le 8223  df-sub 8355  df-neg 8356  df-inn 9147  df-n0 9406  df-z 9483  df-uz 9759

This theorem is referenced by: (None)