Theorem psrbaglesupp 21976

Description: The support of a dominated bag is smaller than the dominating bag. (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
psrbaglesupp	⊢ ((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)

Proof of Theorem psrbaglesupp

Dummy variables 𝑥 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ofr 7663	. . . . . 6 ⊢ ∘r ≤ = {⟨𝑎, 𝑏⟩ ∣ ∀𝑐 ∈ (dom 𝑎 ∩ dom 𝑏)(𝑎‘𝑐) ≤ (𝑏‘𝑐)}
2	1	relopabiv 5795	. . . . 5 ⊢ Rel ∘r ≤
3	2	brrelex1i 5705	. . . 4 ⊢ (𝐺 ∘r ≤ 𝐹 → 𝐺 ∈ V)
4	3	3ad2ant3 1149	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐺 ∈ V)
5		simp2 1151	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐺:𝐼⟶ℕ0)
6		fcdmnn0suppg 12542	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐺:𝐼⟶ℕ0) → (𝐺 supp 0) = (◡𝐺 “ ℕ))
7	4, 5, 6	syl2anc 593	. 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (𝐺 supp 0) = (◡𝐺 “ ℕ))
8		eldifi 4086	. . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ)) → 𝑥 ∈ 𝐼)
9		simp3 1152	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐺 ∘r ≤ 𝐹)
10	5	ffnd 6694	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐺 Fn 𝐼)
11		psrbag.d	. . . . . . . . . . . 12 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
12	11	psrbagf 21972	. . . . . . . . . . 11 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
13	12	3ad2ant1 1147	. . . . . . . . . 10 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐹:𝐼⟶ℕ0)
14	13	ffnd 6694	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐹 Fn 𝐼)
15		simp1 1150	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐹 ∈ 𝐷)
16		inidm 4180	. . . . . . . . 9 ⊢ (𝐼 ∩ 𝐼) = 𝐼
17		eqidd 2765	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) = (𝐺‘𝑥))
18		eqidd 2765	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥))
19	10, 14, 4, 15, 16, 17, 18	ofrfvalg 7670	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (𝐺 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐼 (𝐺‘𝑥) ≤ (𝐹‘𝑥)))
20	9, 19	mpbid 234	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → ∀𝑥 ∈ 𝐼 (𝐺‘𝑥) ≤ (𝐹‘𝑥))
21	20	r19.21bi 3256	. . . . . 6 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ≤ (𝐹‘𝑥))
22	8, 21	sylan2 602	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ≤ (𝐹‘𝑥))
23		fcdmnn0suppg 12542	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
24	15, 13, 23	syl2anc 593	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
25		eqimss 3996	. . . . . . 7 ⊢ ((𝐹 supp 0) = (◡𝐹 “ ℕ) → (𝐹 supp 0) ⊆ (◡𝐹 “ ℕ))
26	24, 25	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (𝐹 supp 0) ⊆ (◡𝐹 “ ℕ))
27		c0ex 11175	. . . . . . 7 ⊢ 0 ∈ V
28	27	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 0 ∈ V)
29	13, 26, 15, 28	suppssrg 8178	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐹‘𝑥) = 0)
30	22, 29	breqtrd 5128	. . . 4 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ≤ 0)
31		ffvelcdm 7064	. . . . . 6 ⊢ ((𝐺:𝐼⟶ℕ0 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ ℕ0)
32	5, 8, 31	syl2an 605	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ∈ ℕ0)
33	32	nn0ge0d 12547	. . . 4 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → 0 ≤ (𝐺‘𝑥))
34	32	nn0red 12545	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ∈ ℝ)
35		0re 11185	. . . . 5 ⊢ 0 ∈ ℝ
36		letri3 11270	. . . . 5 ⊢ (((𝐺‘𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → ((𝐺‘𝑥) = 0 ↔ ((𝐺‘𝑥) ≤ 0 ∧ 0 ≤ (𝐺‘𝑥))))
37	34, 35, 36	sylancl 595	. . . 4 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → ((𝐺‘𝑥) = 0 ↔ ((𝐺‘𝑥) ≤ 0 ∧ 0 ≤ (𝐺‘𝑥))))
38	30, 33, 37	mpbir2and 723	. . 3 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) = 0)
39	5, 38	suppss 8176	. 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (𝐺 supp 0) ⊆ (◡𝐹 “ ℕ))
40	7, 39	eqsstrrd 3973	1 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078  {crab 3416  Vcvv 3456   ∖ cdif 3903   ∩ cin 3905   ⊆ wss 3906   class class class wbr 5102  ^◡ccnv 5648  dom cdm 5649   “ cima 5652  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398   ∘_r cofr 7661   supp csupp 8142   ↑_m cmap 8810  Fincfn 8929  ℝcr 11074  0cc0 11075   ≤ cle 11219  ℕcn 12212  ℕ₀cn0 12483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-ofr 7663  df-om 7849  df-1st 7972  df-2nd 7973  df-supp 8143  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-er 8680  df-map 8812  df-en 8930  df-dom 8931  df-sdom 8932  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-n0 12484

This theorem is referenced by:  psrbaglecl  21977  psrbagcon  21979