Theorem mntoval 31260

Description: Operation value of the monotone function. (Contributed by Thierry Arnoux, 23-Apr-2024.)

Hypotheses

Ref	Expression
mntoval.1	⊢ 𝐴 = (Base‘𝑉)
mntoval.2	⊢ 𝐵 = (Base‘𝑊)
mntoval.3	⊢ ≤ = (le‘𝑉)
mntoval.4	⊢ ≲ = (le‘𝑊)

Assertion

Ref	Expression
mntoval	⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))})

Distinct variable groups:   𝑥,𝐴,𝑦,𝑓   𝐵,𝑓   𝑓,𝑉,𝑥,𝑦   𝑥,𝑊,𝑦,𝑓

Allowed substitution hints:   𝐵(𝑥,𝑦)   ≤ (𝑥,𝑦,𝑓)   𝑋(𝑥,𝑦,𝑓)   𝑌(𝑥,𝑦,𝑓)   ≲ (𝑥,𝑦,𝑓)

Proof of Theorem mntoval

Dummy variables 𝑎 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mnt 31258	. . 3 ⊢ Monot = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌{𝑓 ∈ ((Base‘𝑤) ↑m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦))})
2	1	a1i 11	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → Monot = (𝑣 ∈ V, 𝑤 ∈ V ↦ ⦋(Base‘𝑣) / 𝑎⦌{𝑓 ∈ ((Base‘𝑤) ↑m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦))}))
3		fvexd 6789	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) → (Base‘𝑣) ∈ V)
4		fveq2 6774	. . . . . 6 ⊢ (𝑣 = 𝑉 → (Base‘𝑣) = (Base‘𝑉))
5		mntoval.1	. . . . . 6 ⊢ 𝐴 = (Base‘𝑉)
6	4, 5	eqtr4di 2796	. . . . 5 ⊢ (𝑣 = 𝑉 → (Base‘𝑣) = 𝐴)
7	6	adantr 481	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) → (Base‘𝑣) = 𝐴)
8		simplr 766	. . . . . . . 8 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑤 = 𝑊)
9	8	fveq2d 6778	. . . . . . 7 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
10		mntoval.2	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
11	9, 10	eqtr4di 2796	. . . . . 6 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
12		simpr 485	. . . . . 6 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
13	11, 12	oveq12d 7293	. . . . 5 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → ((Base‘𝑤) ↑m 𝑎) = (𝐵 ↑m 𝐴))
14		simpll 764	. . . . . . . . . . 11 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑣 = 𝑉)
15	14	fveq2d 6778	. . . . . . . . . 10 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑣) = (le‘𝑉))
16		mntoval.3	. . . . . . . . . 10 ⊢ ≤ = (le‘𝑉)
17	15, 16	eqtr4di 2796	. . . . . . . . 9 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑣) = ≤ )
18	17	breqd 5085	. . . . . . . 8 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (𝑥(le‘𝑣)𝑦 ↔ 𝑥 ≤ 𝑦))
19	8	fveq2d 6778	. . . . . . . . . 10 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑤) = (le‘𝑊))
20		mntoval.4	. . . . . . . . . 10 ⊢ ≲ = (le‘𝑊)
21	19, 20	eqtr4di 2796	. . . . . . . . 9 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑤) = ≲ )
22	21	breqd 5085	. . . . . . . 8 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → ((𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦) ↔ (𝑓‘𝑥) ≲ (𝑓‘𝑦)))
23	18, 22	imbi12d 345	. . . . . . 7 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → ((𝑥(le‘𝑣)𝑦 → (𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦)) ↔ (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))))
24	12, 23	raleqbidv 3336	. . . . . 6 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (∀𝑦 ∈ 𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦)) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))))
25	12, 24	raleqbidv 3336	. . . . 5 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))))
26	13, 25	rabeqbidv 3420	. . . 4 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → {𝑓 ∈ ((Base‘𝑤) ↑m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦))} = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))})
27	3, 7, 26	csbied2 3872	. . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) → ⦋(Base‘𝑣) / 𝑎⦌{𝑓 ∈ ((Base‘𝑤) ↑m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦))} = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))})
28	27	adantl 482	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ (𝑣 = 𝑉 ∧ 𝑤 = 𝑊)) → ⦋(Base‘𝑣) / 𝑎⦌{𝑓 ∈ ((Base‘𝑤) ↑m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦))} = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))})
29		elex 3450	. . 3 ⊢ (𝑉 ∈ 𝑋 → 𝑉 ∈ V)
30	29	adantr 481	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → 𝑉 ∈ V)
31		elex 3450	. . 3 ⊢ (𝑊 ∈ 𝑌 → 𝑊 ∈ V)
32	31	adantl 482	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → 𝑊 ∈ V)
33		eqid 2738	. . 3 ⊢ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))} = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))}
34		ovexd 7310	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐵 ↑m 𝐴) ∈ V)
35	33, 34	rabexd 5257	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))} ∈ V)
36	2, 28, 30, 32, 35	ovmpod 7425	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  Vcvv 3432  ⦋csb 3832   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277   ↑_m cmap 8615  Basecbs 16912  lecple 16969  Monotcmnt 31256

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-mnt 31258

This theorem is referenced by:  ismnt  31261