Theorem psrbaglecl 15060

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 1029	. 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺:𝐼⟶ℕ0)
2		simp1 1028	. . . . 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 15054	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
6	5	ffnd 5534	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼)
7	3, 6	fndmexd 5581	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → 𝐼 ∈ V)
8	7	3ad2ant1 1049	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐼 ∈ V)
9	4	psrbag 15053	. . . . . 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 15058	. . 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 1049	. . . . . . . . . 10 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐹:𝐼⟶ℕ0)
16		ffn 5533	. . . . . . . . . 10 ⊢ (𝐹:𝐼⟶ℕ0 → 𝐹 Fn 𝐼)
17		elpreima 5828	. . . . . . . . . 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 5844	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (◡𝐹 “ ℕ)) → (𝐺‘𝑥) ∈ ℕ0)
21	20	nn0zd 9766	. . . . . 6 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (◡𝐹 “ ℕ)) → (𝐺‘𝑥) ∈ ℤ)
22		elnndc 10012	. . . . . 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 5533	. . . . . . . . 9 ⊢ (𝐺:𝐼⟶ℕ0 → 𝐺 Fn 𝐼)
25		elpreima 5828	. . . . . . . . 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 445	. . . . . 6 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (◡𝐹 “ ℕ)) → (𝑥 ∈ (◡𝐺 “ ℕ) ↔ (𝐺‘𝑥) ∈ ℕ))
29	28	dcbid 850	. . . . 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 2623	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ∀𝑥 ∈ (◡𝐹 “ ℕ)DECID 𝑥 ∈ (◡𝐺 “ ℕ))
32		ssfidc 7245	. . 3 ⊢ (((◡𝐹 “ ℕ) ∈ Fin ∧ (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ) ∧ ∀𝑥 ∈ (◡𝐹 “ ℕ)DECID 𝑥 ∈ (◡𝐺 “ ℕ)) → (◡𝐺 “ ℕ) ∈ Fin)
33	12, 13, 31, 32	syl3anc 1278	. 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (◡𝐺 “ ℕ) ∈ Fin)
34	4	psrbag 15053	. . 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 957	1 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺 ∈ 𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821   ⊆ wss 3220   class class class wbr 4130  ^◡ccnv 4773   “ cima 4777   Fn wfn 5372  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085   ∘_𝑟 cofr 6301   ↑_𝑚 cmap 6922  Fincfn 7022   ≤ cle 8361  ℕcn 9304  ℕ₀cn0 9563  ℤcz 9644

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-ofr 6303  df-1o 6687  df-er 6807  df-map 6924  df-en 7023  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922

This theorem is used by: (None)