Theorem psrbaglecl 14771

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 1025	. 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺:𝐼⟶ℕ0)
2		simp1 1024	. . . . 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 14766	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
6	5	ffnd 5490	. . . . . . . 8 ⊢ (𝐹 ∈ 𝐷 → 𝐹 Fn 𝐼)
7	3, 6	fndmexd 5534	. . . . . . 7 ⊢ (𝐹 ∈ 𝐷 → 𝐼 ∈ V)
8	7	3ad2ant1 1045	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐼 ∈ V)
9	4	psrbag 14765	. . . . . 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 14769	. . 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 1045	. . . . . . . . . 10 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐹:𝐼⟶ℕ0)
16		ffn 5489	. . . . . . . . . 10 ⊢ (𝐹:𝐼⟶ℕ0 → 𝐹 Fn 𝐼)
17		elpreima 5775	. . . . . . . . . 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 5791	. . . . . . 7 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (◡𝐹 “ ℕ)) → (𝐺‘𝑥) ∈ ℕ0)
21	20	nn0zd 9661	. . . . . 6 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (◡𝐹 “ ℕ)) → (𝐺‘𝑥) ∈ ℤ)
22		elnndc 9907	. . . . . 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 5489	. . . . . . . . 9 ⊢ (𝐺:𝐼⟶ℕ0 → 𝐺 Fn 𝐼)
25		elpreima 5775	. . . . . . . . 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 846	. . . . 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 2606	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → ∀𝑥 ∈ (◡𝐹 “ ℕ)DECID 𝑥 ∈ (◡𝐺 “ ℕ))
32		ssfidc 7173	. . 3 ⊢ (((◡𝐹 “ ℕ) ∈ Fin ∧ (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ) ∧ ∀𝑥 ∈ (◡𝐹 “ ℕ)DECID 𝑥 ∈ (◡𝐺 “ ℕ)) → (◡𝐺 “ ℕ) ∈ Fin)
33	12, 13, 31, 32	syl3anc 1274	. 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → (◡𝐺 “ ℕ) ∈ Fin)
34	4	psrbag 14765	. . 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 953	1 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → 𝐺 ∈ 𝐷)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 842   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  ∀wral 2511  {crab 2515  Vcvv 2803   ⊆ wss 3201   class class class wbr 4093  ^◡ccnv 4730   “ cima 4734   Fn wfn 5328  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028   ∘_𝑟 cofr 6243   ↑_𝑚 cmap 6860  Fincfn 6952   ≤ cle 8274  ℕcn 9202  ℕ₀cn0 9461  ℤcz 9540

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-ofr 6245  df-1o 6625  df-er 6745  df-map 6862  df-en 6953  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817

This theorem is referenced by: (None)