Theorem mntoval 33543

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 33541	. . 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 6900	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) → (Base‘𝑣) ∈ V)
4		fveq2 6885	. . . . . 6 ⊢ (𝑣 = 𝑉 → (Base‘𝑣) = (Base‘𝑉))
5		mntoval.1	. . . . . 6 ⊢ 𝐴 = (Base‘𝑉)
6	4, 5	eqtr4di 2814	. . . . 5 ⊢ (𝑣 = 𝑉 → (Base‘𝑣) = 𝐴)
7	6	adantr 486	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) → (Base‘𝑣) = 𝐴)
8		simplr 781	. . . . . . . 8 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑤 = 𝑊)
9	8	fveq2d 6889	. . . . . . 7 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
10		mntoval.2	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
11	9, 10	eqtr4di 2814	. . . . . 6 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
12		simpr 490	. . . . . 6 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
13	11, 12	oveq12d 7438	. . . . 5 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → ((Base‘𝑤) ↑m 𝑎) = (𝐵 ↑m 𝐴))
14		simpll 779	. . . . . . . . . . 11 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑣 = 𝑉)
15	14	fveq2d 6889	. . . . . . . . . 10 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑣) = (le‘𝑉))
16		mntoval.3	. . . . . . . . . 10 ⊢ ≤ = (le‘𝑉)
17	15, 16	eqtr4di 2814	. . . . . . . . 9 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑣) = ≤ )
18	17	breqd 5114	. . . . . . . 8 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (𝑥(le‘𝑣)𝑦 ↔ 𝑥 ≤ 𝑦))
19	8	fveq2d 6889	. . . . . . . . . 10 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑤) = (le‘𝑊))
20		mntoval.4	. . . . . . . . . 10 ⊢ ≲ = (le‘𝑊)
21	19, 20	eqtr4di 2814	. . . . . . . . 9 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑤) = ≲ )
22	21	breqd 5114	. . . . . . . 8 ⊢ (((𝑣 = 𝑉 ∧ 𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → ((𝑓‘𝑥)(le‘𝑤)(𝑓‘𝑦) ↔ (𝑓‘𝑥) ≲ (𝑓‘𝑦)))
23	18, 22	imbi12d 347	. . . . . . 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 3335	. . . . . 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 3335	. . . . 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 3430	. . . 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 3884	. . 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 487	. 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 3472	. . 3 ⊢ (𝑉 ∈ 𝑋 → 𝑉 ∈ V)
30	29	adantr 486	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → 𝑉 ∈ V)
31		elex 3472	. . 3 ⊢ (𝑊 ∈ 𝑌 → 𝑊 ∈ V)
32	31	adantl 487	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → 𝑊 ∈ V)
33		eqid 2761	. . 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 7455	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐵 ↑m 𝐴) ∈ V)
35	33, 34	rabexd 5301	. 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 7572	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413  Vcvv 3451  ⦋csb 3847   class class class wbr 5103  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422   ↑_m cmap 8847  Basecbs 17387  lecple 17435  Monotcmnt 33539

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-mnt 33541

This theorem is used by:  ismnt  33544