Theorem psrbaglesuppg 14676

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 1063	. . . . . . . . 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 14673	. . . . . . . . . 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 1064	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐺:𝐼⟶ℕ0)
10	9	ffnd 5480	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐺 Fn 𝐼)
11		elpreima 5762	. . . . . . . . . 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 5779	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹‘𝑥) ∈ ℕ0)
16	15	nn0zd 9590	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹‘𝑥) ∈ ℤ)
17		1red 8184	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 1 ∈ ℝ)
18		ffun 5482	. . . . . . . . . 10 ⊢ (𝐺:𝐼⟶ℕ0 → Fun 𝐺)
19	18	3ad2ant2 1043	. . . . . . . . 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 5757	. . . . . . . 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 9146	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐺‘𝑥) ∈ ℝ)
24	15	nn0red 9446	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → (𝐹‘𝑥) ∈ ℝ)
25	22	nnge1d 9176	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 1 ≤ (𝐺‘𝑥))
26		simplr3 1065	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐺 ∘𝑟 ≤ 𝐹)
27	7	ffnd 5480	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝐹 Fn 𝐼)
28		inidm 3414	. . . . . . . 8 ⊢ (𝐼 ∩ 𝐼) = 𝐼
29		eqidd 2230	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) = (𝐺‘𝑥))
30		eqidd 2230	. . . . . . . 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 6241	. . . . . . 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 1373	. . . . . 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 8293	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 1 ≤ (𝐹‘𝑥))
34		elnnz1 9492	. . . . 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 5483	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → Fun 𝐹)
37	7	fdmd 5486	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → dom 𝐹 = 𝐼)
38	14, 37	eleqtrrd 2309	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) ∧ 𝑥 ∈ (◡𝐺 “ ℕ)) → 𝑥 ∈ dom 𝐹)
39		fvimacnv 5758	. . . . 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 3231	1 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘𝑟 ≤ 𝐹)) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  {crab 2512   ⊆ wss 3198   class class class wbr 4086  ^◡ccnv 4722  dom cdm 4723   “ cima 4726  Fun wfun 5318   Fn wfn 5319  ⟶wf 5320  ‘cfv 5324  (class class class)co 6013   ∘_𝑟 cofr 6229   ↑_𝑚 cmap 6812  Fincfn 6904  1c1 8023   ≤ cle 8205  ℕcn 9133  ℕ₀cn0 9392  ℤcz 9469

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-ofr 6231  df-map 6814  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470

This theorem is referenced by: (None)