Theorem psrbaglesuppg 14405

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 14402	. . . . . . . . . 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 5425	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐺 Fn 𝐼)
11		elpreima 5698	. . . . . . . . . 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 5715	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹‘𝑥) ∈ ℕ0)
16	15	nn0zd 9492	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹‘𝑥) ∈ ℤ)
17		1red 8086	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 1 ∈ ℝ)
18		ffun 5427	. . . . . . . . . 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 5693	. . . . . . . 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 9048	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐺‘𝑥) ∈ ℝ)
24	15	nn0red 9348	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹‘𝑥) ∈ ℝ)
25	22	nnge1d 9078	. . . . . 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 5425	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐹 Fn 𝐼)
28		inidm 3381	. . . . . . . 8 ⊢ (𝐼 ∩ 𝐼) = 𝐼
29		eqidd 2205	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) = (𝐺‘𝑥))
30		eqidd 2205	. . . . . . . 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 6168	. . . . . . 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 1351	. . . . . 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 8195	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 1 ≤ (𝐹‘𝑥))
34		elnnz1 9394	. . . . 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 5428	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → Fun 𝐹)
37	7	fdmd 5431	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → dom 𝐹 = 𝐼)
38	14, 37	eleqtrrd 2284	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝑥 ∈ dom 𝐹)
39		fvimacnv 5694	. . . . 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 3198	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 1372   ∈ wcel 2175  {crab 2487   ⊆ wss 3165   class class class wbr 4043  ^◡ccnv 4673  dom cdm 4674   “ cima 4677  Fun wfun 5264   Fn wfn 5265  ⟶wf 5266  ‘cfv 5270  (class class class)co 5943   ∘_𝑟 cofr 6156   ↑_𝑚 cmap 6734  Fincfn 6826  1c1 7925   ≤ cle 8107  ℕcn 9035  ℕ₀cn0 9294  ℤcz 9371

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-ofr 6158  df-map 6736  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-inn 9036  df-n0 9295  df-z 9372

This theorem is referenced by: (None)