Theorem psrbaglesuppg 14821

Description: The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypothesis

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

Assertion

Ref	Expression
psrbaglesuppg	⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))

Distinct variable groups:   𝑓,𝐹   𝑓,𝐼

Allowed substitution hints:   𝐷(𝑓)   𝐺(𝑓)   𝑉(𝑓)

Proof of Theorem psrbaglesuppg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr1 1066	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐹 ∈ 𝐷)
2		simpll 527	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐼 ∈ 𝑉)
3		psrbag.d	. . . . . . . . . . 11 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
4	3	psrbag 14817	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin)))
5	2, 4	syl 14	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹 ∈ 𝐷 ↔ (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin)))
6	1, 5	mpbid 147	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹:𝐼⟶ℕ0 ∧ (◡𝐹 “ ℕ) ∈ Fin))
7	6	simpld 112	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐹:𝐼⟶ℕ0)
8		simpr 110	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝑥 ∈ (◡𝐺 “ ℕ))
9		simplr2 1067	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐺:𝐼⟶ℕ0)
10	9	ffnd 5509	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐺 Fn 𝐼)
11		elpreima 5797	. . . . . . . . . 10 ⊢ (𝐺 Fn 𝐼 → (𝑥 ∈ (◡𝐺 “ ℕ) ↔ (𝑥 ∈ 𝐼 ∧ (𝐺‘𝑥) ∈ ℕ)))
12	10, 11	syl 14	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝑥 ∈ (◡𝐺 “ ℕ) ↔ (𝑥 ∈ 𝐼 ∧ (𝐺‘𝑥) ∈ ℕ)))
13	8, 12	mpbid 147	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝑥 ∈ 𝐼 ∧ (𝐺‘𝑥) ∈ ℕ))
14	13	simpld 112	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝑥 ∈ 𝐼)
15	7, 14	ffvelcdmd 5813	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹‘𝑥) ∈ ℕ0)
16	15	nn0zd 9698	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹‘𝑥) ∈ ℤ)
17		1red 8289	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 1 ∈ ℝ)
18		ffun 5511	. . . . . . . . . 10 ⊢ (𝐺:𝐼⟶ℕ0 → Fun 𝐺)
19	18	3ad2ant2 1046	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹) → Fun 𝐺)
20	19	ad2antlr 489	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → Fun 𝐺)
21		fvimacnvi 5792	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐺‘𝑥) ∈ ℕ)
22	20, 8, 21	syl2anc 411	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐺‘𝑥) ∈ ℕ)
23	22	nnred 9250	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐺‘𝑥) ∈ ℝ)
24	15	nn0red 9554	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹‘𝑥) ∈ ℝ)
25	22	nnge1d 9280	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 1 ≤ (𝐺‘𝑥))
26		simplr3 1068	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐺 ∘𝑟 ≤ 𝐹)
27	7	ffnd 5509	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐹 Fn 𝐼)
28		inidm 3430	. . . . . . . 8 ⊢ (𝐼 ∩ 𝐼) = 𝐼
29		eqidd 2233	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) = (𝐺‘𝑥))
30		eqidd 2233	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥))
31	10, 27, 2, 2, 28, 29, 30	ofrval 6277	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) ∧ 𝐺 ∘𝑟 ≤ 𝐹 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ≤ (𝐹‘𝑥))
32	26, 14, 31	mpd3an23 1376	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐺‘𝑥) ≤ (𝐹‘𝑥))
33	17, 23, 24, 25, 32	letrd 8397	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 1 ≤ (𝐹‘𝑥))
34		elnnz1 9600	. . . . 5 ⊢ ((𝐹‘𝑥) ∈ ℕ ↔ ((𝐹‘𝑥) ∈ ℤ ∧ 1 ≤ (𝐹‘𝑥)))
35	16, 33, 34	sylanbrc 417	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹‘𝑥) ∈ ℕ)
36	7	ffund 5512	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → Fun 𝐹)
37	7	fdmd 5515	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → dom 𝐹 = 𝐼)
38	14, 37	eleqtrrd 2312	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝑥 ∈ dom 𝐹)
39		fvimacnv 5793	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ ℕ ↔ 𝑥 ∈ (◡𝐹 “ ℕ)))
40	36, 38, 39	syl2anc 411	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → ((𝐹‘𝑥) ∈ ℕ ↔ 𝑥 ∈ (◡𝐹 “ ℕ)))
41	35, 40	mpbid 147	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝑥 ∈ (◡𝐹 “ ℕ))
42	41	ex 115	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → (𝑥 ∈ (◡𝐺 “ ℕ) → 𝑥 ∈ (◡𝐹 “ ℕ)))
43	42	ssrdv 3244	1 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  {crab 2524   ⊆ wss 3211   class class class wbr 4109  ^◡ccnv 4748  dom cdm 4749   “ cima 4752  Fun wfun 5346   Fn wfn 5347  ⟶wf 5348  ‘cfv 5352  (class class class)co 6050   ∘_𝑟 cofr 6265   ↑_𝑚 cmap 6882  Fincfn 6975  1c1 8128   ≤ cle 8309  ℕcn 9237  ℕ₀cn0 9496  ℤcz 9577

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-ofr 6267  df-map 6884  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578

This theorem is referenced by:  psrbaglesupp  14822