Theorem mntoval 32924

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 32922	. . 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 6837	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) → (Base‘𝑣) ∈ V)
4		fveq2 6822	. . . . . 6 ⊢ (𝑣 = 𝑉 → (Base‘𝑣) = (Base‘𝑉))
5		mntoval.1	. . . . . 6 ⊢ 𝐴 = (Base‘𝑉)
6	4, 5	eqtr4di 2782	. . . . 5 ⊢ (𝑣 = 𝑉 → (Base‘𝑣) = 𝐴)
7	6	adantr 480	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) → (Base‘𝑣) = 𝐴)
8		simplr 768	. . . . . . . 8 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑤 = 𝑊)
9	8	fveq2d 6826	. . . . . . 7 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
10		mntoval.2	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
11	9, 10	eqtr4di 2782	. . . . . 6 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
12		simpr 484	. . . . . 6 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
13	11, 12	oveq12d 7367	. . . . 5 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → ((Base‘𝑤) ↑m 𝑎) = (𝐵 ↑m 𝐴))
14		simpll 766	. . . . . . . . . . 11 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑣 = 𝑉)
15	14	fveq2d 6826	. . . . . . . . . 10 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑣) = (le‘𝑉))
16		mntoval.3	. . . . . . . . . 10 ⊢ ≤ = (le‘𝑉)
17	15, 16	eqtr4di 2782	. . . . . . . . 9 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑣) = ≤ )
18	17	breqd 5103	. . . . . . . 8 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (𝑥(le‘𝑣)𝑦 ↔ 𝑥 ≤ 𝑦))
19	8	fveq2d 6826	. . . . . . . . . 10 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑤) = (le‘𝑊))
20		mntoval.4	. . . . . . . . . 10 ⊢ ≲ = (le‘𝑊)
21	19, 20	eqtr4di 2782	. . . . . . . . 9 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑤) = ≲ )
22	21	breqd 5103	. . . . . . . 8 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → ((𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦) ↔ (𝑓‘𝑥) ≲ (𝑓‘𝑦)))
23	18, 22	imbi12d 344	. . . . . . 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 3309	. . . . . 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 3309	. . . . 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 3413	. . . 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 3888	. . 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 481	. 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 3457	. . 3 ⊢ (𝑉 ∈ 𝑋 → 𝑉 ∈ V)
30	29	adantr 480	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → 𝑉 ∈ V)
31		elex 3457	. . 3 ⊢ (𝑊 ∈ 𝑌 → 𝑊 ∈ V)
32	31	adantl 481	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → 𝑊 ∈ V)
33		eqid 2729	. . 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 7384	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐵 ↑m 𝐴) ∈ V)
35	33, 34	rabexd 5279	. 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 7501	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 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3394  Vcvv 3436  ⦋csb 3851   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351   ↑_m cmap 8753  Basecbs 17120  lecple 17168  Monotcmnt 32920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-mnt 32922

This theorem is referenced by:  ismnt  32925