Theorem psrbaglesupp 21469

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,𝐺   𝑓,𝐼

Allowed substitution hint:   𝐷(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 7668	. . . . . 6 ⊢ ∘r ≤ = {⟨𝑎, 𝑏⟩ ∣ ∀𝑐 ∈ (dom 𝑎 ∩ dom 𝑏)(𝑎‘𝑐) ≤ (𝑏‘𝑐)}
2	1	relopabiv 5819	. . . . 5 ⊢ Rel ∘r ≤
3	2	brrelex1i 5731	. . . 4 ⊢ (𝐺 ∘r ≤ 𝐹 → 𝐺 ∈ V)
4	3	3ad2ant3 1136	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐺 ∈ V)
5		simp2 1138	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐺:𝐼⟶ℕ0)
6		fcdmnn0suppg 12527	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐺:𝐼⟶ℕ0) → (𝐺 supp 0) = (◡𝐺 “ ℕ))
7	4, 5, 6	syl2anc 585	. 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (𝐺 supp 0) = (◡𝐺 “ ℕ))
8		eldifi 4126	. . . . . 6 ⊢ (𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ)) → 𝑥 ∈ 𝐼)
9		simp3 1139	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐺 ∘r ≤ 𝐹)
10	5	ffnd 6716	. . . . . . . . 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 21463	. . . . . . . . . . 11 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
13	12	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐹:𝐼⟶ℕ0)
14	13	ffnd 6716	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐹 Fn 𝐼)
15		simp1 1137	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐹 ∈ 𝐷)
16		inidm 4218	. . . . . . . . 9 ⊢ (𝐼 ∩ 𝐼) = 𝐼
17		eqidd 2734	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) = (𝐺‘𝑥))
18		eqidd 2734	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥))
19	10, 14, 4, 15, 16, 17, 18	ofrfvalg 7675	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (𝐺 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐼 (𝐺‘𝑥) ≤ (𝐹‘𝑥)))
20	9, 19	mpbid 231	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → ∀𝑥 ∈ 𝐼 (𝐺‘𝑥) ≤ (𝐹‘𝑥))
21	20	r19.21bi 3249	. . . . . 6 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ≤ (𝐹‘𝑥))
22	8, 21	sylan2 594	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ≤ (𝐹‘𝑥))
23		fcdmnn0suppg 12527	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
24	15, 13, 23	syl2anc 585	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
25		eqimss 4040	. . . . . . 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 11205	. . . . . . 7 ⊢ 0 ∈ V
28	27	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 0 ∈ V)
29	13, 26, 15, 28	suppssrg 8179	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐹‘𝑥) = 0)
30	22, 29	breqtrd 5174	. . . 4 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ≤ 0)
31		ffvelcdm 7081	. . . . . 6 ⊢ ((𝐺:𝐼⟶ℕ0 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ ℕ0)
32	5, 8, 31	syl2an 597	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ∈ ℕ0)
33	32	nn0ge0d 12532	. . . 4 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → 0 ≤ (𝐺‘𝑥))
34	32	nn0red 12530	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ∈ ℝ)
35		0re 11213	. . . . 5 ⊢ 0 ∈ ℝ
36		letri3 11296	. . . . 5 ⊢ (((𝐺‘𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → ((𝐺‘𝑥) = 0 ↔ ((𝐺‘𝑥) ≤ 0 ∧ 0 ≤ (𝐺‘𝑥))))
37	34, 35, 36	sylancl 587	. . . 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 712	. . 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 4021	1 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433  Vcvv 3475   ∖ cdif 3945   ∩ cin 3947   ⊆ wss 3948   class class class wbr 5148  ^◡ccnv 5675  dom cdm 5676   “ cima 5679  ⟶wf 6537  ‘cfv 6541  (class class class)co 7406   ∘_r cofr 7666   supp csupp 8143   ↑_m cmap 8817  Fincfn 8936  ℝcr 11106  0cc0 11107   ≤ cle 11246  ℕcn 12209  ℕ₀cn0 12469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-ofr 7668  df-om 7853  df-1st 7972  df-2nd 7973  df-supp 8144  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-n0 12470

This theorem is referenced by:  psrbaglecl  21471  psrbagcon  21475