Theorem psrbaglesupp 21326

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 7618	. . . . . 6 ⊢ ∘r ≤ = {⟨𝑎, 𝑏⟩ ∣ ∀𝑐 ∈ (dom 𝑎 ∩ dom 𝑏)(𝑎‘𝑐) ≤ (𝑏‘𝑐)}
2	1	relopabiv 5776	. . . . 5 ⊢ Rel ∘r ≤
3	2	brrelex1i 5688	. . . 4 ⊢ (𝐺 ∘r ≤ 𝐹 → 𝐺 ∈ V)
4	3	3ad2ant3 1135	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐺 ∈ V)
5		simp2 1137	. . 3 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐺:𝐼⟶ℕ0)
6		fcdmnn0suppg 12471	. . 3 ⊢ ((𝐺 ∈ V ∧ 𝐺:𝐼⟶ℕ0) → (𝐺 supp 0) = (◡𝐺 “ ℕ))
7	4, 5, 6	syl2anc 584	. 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 1138	. . . . . . . 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 21320	. . . . . . . . . . 11 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0)
13	12	3ad2ant1 1133	. . . . . . . . . 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 1136	. . . . . . . . 9 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 𝐹 ∈ 𝐷)
16		inidm 4178	. . . . . . . . 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 7625	. . . . . . . 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 3234	. . . . . 6 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ≤ (𝐹‘𝑥))
22	8, 21	sylan2 593	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ≤ (𝐹‘𝑥))
23		fcdmnn0suppg 12471	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
24	15, 13, 23	syl2anc 584	. . . . . . 7 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (𝐹 supp 0) = (◡𝐹 “ ℕ))
25		eqimss 4000	. . . . . . 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 11149	. . . . . . 7 ⊢ 0 ∈ V
28	27	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → 0 ∈ V)
29	13, 26, 15, 28	suppssrg 8128	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐹‘𝑥) = 0)
30	22, 29	breqtrd 5131	. . . 4 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ≤ 0)
31		ffvelcdm 7032	. . . . . 6 ⊢ ((𝐺:𝐼⟶ℕ0 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ ℕ0)
32	5, 8, 31	syl2an 596	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ∈ ℕ0)
33	32	nn0ge0d 12476	. . . 4 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → 0 ≤ (𝐺‘𝑥))
34	32	nn0red 12474	. . . . 5 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) ∈ ℝ)
35		0re 11157	. . . . 5 ⊢ 0 ∈ ℝ
36		letri3 11240	. . . . 5 ⊢ (((𝐺‘𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → ((𝐺‘𝑥) = 0 ↔ ((𝐺‘𝑥) ≤ 0 ∧ 0 ≤ (𝐺‘𝑥))))
37	34, 35, 36	sylancl 586	. . . 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 711	. . 3 ⊢ (((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) ∧ 𝑥 ∈ (𝐼 ∖ (◡𝐹 “ ℕ))) → (𝐺‘𝑥) = 0)
39	5, 38	suppss 8125	. 2 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (𝐺 supp 0) ⊆ (◡𝐹 “ ℕ))
40	7, 39	eqsstrrd 3983	1 ⊢ ((𝐹 ∈ 𝐷 ∧ 𝐺:𝐼⟶ℕ0 ∧ 𝐺 ∘r ≤ 𝐹) → (◡𝐺 “ ℕ) ⊆ (◡𝐹 “ ℕ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  {crab 3407  Vcvv 3445   ∖ cdif 3907   ∩ cin 3909   ⊆ wss 3910   class class class wbr 5105  ^◡ccnv 5632  dom cdm 5633   “ cima 5636  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ∘_r cofr 7616   supp csupp 8092   ↑_m cmap 8765  Fincfn 8883  ℝcr 11050  0cc0 11051   ≤ cle 11190  ℕcn 12153  ℕ₀cn0 12413

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414

This theorem is referenced by:  psrbaglecl  21328  psrbagcon  21332