Theorem psrbaglesuppg 14226

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 1041	. . . . . . . . 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 14223	. . . . . . . . . 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 1042	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐺:𝐼⟶ℕ0)
10	9	ffnd 5408	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐺 Fn 𝐼)
11		elpreima 5681	. . . . . . . . . 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 5698	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹‘𝑥) ∈ ℕ0)
16	15	nn0zd 9446	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹‘𝑥) ∈ ℤ)
17		1red 8041	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 1 ∈ ℝ)
18		ffun 5410	. . . . . . . . . 10 ⊢ (𝐺:𝐼⟶ℕ0 → Fun 𝐺)
19	18	3ad2ant2 1021	. . . . . . . . 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 5676	. . . . . . . 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 9003	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐺‘𝑥) ∈ ℝ)
24	15	nn0red 9303	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹‘𝑥) ∈ ℝ)
25	22	nnge1d 9033	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 1 ≤ (𝐺‘𝑥))
26		simplr3 1043	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐺 ∘𝑟 ≤ 𝐹)
27	7	ffnd 5408	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐹 Fn 𝐼)
28		inidm 3372	. . . . . . . 8 ⊢ (𝐼 ∩ 𝐼) = 𝐼
29		eqidd 2197	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) = (𝐺‘𝑥))
30		eqidd 2197	. . . . . . . 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 6146	. . . . . . 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 1350	. . . . . 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 8150	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 1 ≤ (𝐹‘𝑥))
34		elnnz1 9349	. . . . 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 5411	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → Fun 𝐹)
37	7	fdmd 5414	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → dom 𝐹 = 𝐼)
38	14, 37	eleqtrrd 2276	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝑥 ∈ dom 𝐹)
39		fvimacnv 5677	. . . . 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 3189	1 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  {crab 2479   ⊆ wss 3157   class class class wbr 4033  ^◡ccnv 4662  dom cdm 4663   “ cima 4666  Fun wfun 5252   Fn wfn 5253  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922   ∘_𝑟 cofr 6134   ↑_𝑚 cmap 6707  Fincfn 6799  1c1 7880   ≤ cle 8062  ℕcn 8990  ℕ₀cn0 9249  ℤcz 9326

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-ofr 6136  df-map 6709  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327

This theorem is referenced by: (None)