Theorem ismnt 33243

Description: Express the statement "𝐹 is monotone". (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
ismnt	⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦   𝑥,𝐹,𝑦

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

Proof of Theorem ismnt

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mntoval.1	. . . . 5 ⊢ 𝐴 = (Base‘𝑉)
2		mntoval.2	. . . . 5 ⊢ 𝐵 = (Base‘𝑊)
3		mntoval.3	. . . . 5 ⊢ ≤ = (le‘𝑉)
4		mntoval.4	. . . . 5 ⊢ ≲ = (le‘𝑊)
5	1, 2, 3, 4	mntoval 33242	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))})
6	5	eleq2d 2855	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ 𝐹 ∈ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))}))
7		fveq1 6881	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
8		fveq1 6881	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
9	7, 8	breq12d 5126	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ≲ (𝑓‘𝑦) ↔ (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
10	9	imbi2d 343	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦)) ↔ (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))))
11	10	2ralbidv 3235	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))))
12	11	elrab 3659	. . 3 ⊢ (𝐹 ∈ {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))))
13	6, 12	bitrdi 290	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))))
14	2	fvexi 6896	. . . 4 ⊢ 𝐵 ∈ V
15	1	fvexi 6896	. . . 4 ⊢ 𝐴 ∈ V
16	14, 15	elmap 8868	. . 3 ⊢ (𝐹 ∈ (𝐵 ↑m 𝐴) ↔ 𝐹:𝐴⟶𝐵)
17	16	anbi1i 635	. 2 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))))
18	13, 17	bitrdi 290	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423   class class class wbr 5113  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ↑_m cmap 8823  Basecbs 17268  lecple 17316  Monotcmnt 33238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8825  df-mnt 33240

This theorem is referenced by:  ismntd  33244  mntf  33245  mgcmnt1d  33257  mgcmnt2d  33258