Theorem psrbaglesupp 21902

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 7632	. . . . . 6 ⊢ ∘r ≤ = {⟨𝑎, 𝑏⟩ ∣ ∀𝑐 ∈ (dom 𝑎 ∩ dom 𝑏)(𝑎‘𝑐) ≤ (𝑏‘𝑐)}
2	1	relopabiv 5776	. . . . 5 ⊢ Rel ∘r ≤
3	2	brrelex1i 5687	. . . 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 12496	. . 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 4071	. . . . . 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 6669	. . . . . . . . 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 21898	. . . . . . . . . . 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 6669	. . . . . . . . 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 4167	. . . . . . . . 9 ⊢ (𝐼 ∩ 𝐼) = 𝐼
17		eqidd 2737	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) = (𝐺‘𝑥))
18		eqidd 2737	. . . . . . . . 9 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) = (𝐹‘𝑥))
19	10, 14, 4, 15, 16, 17, 18	ofrfvalg 7639	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (𝐺 ∘r ≤ 𝐹 ↔ ∀𝑥 ∈ 𝐼 (𝐺‘𝑥) ≤ (𝐹‘𝑥)))
20	9, 19	mpbid 232	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → ∀𝑥 ∈ 𝐼 (𝐺‘𝑥) ≤ (𝐹‘𝑥))
21	20	r19.21bi 3229	. . . . . 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 12496	. . . . . . . 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 3980	. . . . . . 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 11138	. . . . . . 7 ⊢ 0 ∈ V
28	27	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 0 ∈ V)
29	13, 26, 15, 28	suppssrg 8146	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐹‘𝑥) = 0)
30	22, 29	breqtrd 5111	. . . 4 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ≤ 0)
31		ffvelcdm 7033	. . . . . 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 12501	. . . 4 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → 0 ≤ (𝐺‘𝑥))
34	32	nn0red 12499	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ∈ ℝ)
35		0re 11146	. . . . 5 ⊢ 0 ∈ ℝ
36		letri3 11231	. . . . 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 714	. . 3 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) = 0)
39	5, 38	suppss 8144	. 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (𝐺 supp 0) ⊆ (◡𝐹 “ ℕ))
40	7, 39	eqsstrrd 3957	1 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389  Vcvv 3429   ∖ cdif 3886   ∩ cin 3888   ⊆ wss 3889   class class class wbr 5085  ^◡ccnv 5630  dom cdm 5631   “ cima 5634  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∘_r cofr 7630   supp csupp 8110   ↑_m cmap 8773  Fincfn 8893  ℝcr 11037  0cc0 11038   ≤ cle 11180  ℕcn 12174  ℕ₀cn0 12437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-ofr 7632  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438

This theorem is referenced by:  psrbaglecl  21903  psrbagcon  21905