Theorem ismnt 33303

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 33302	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))})
6	5	eleq2d 2849	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ 𝐹 ∈ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))}))
7		fveq1 6880	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
8		fveq1 6880	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
9	7, 8	breq12d 5122	. . . . . 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 3229	. . . 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 3650	. . 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 6895	. . . 4 ⊢ 𝐵 ∈ V
15	1	fvexi 6895	. . . 4 ⊢ 𝐴 ∈ V
16	14, 15	elmap 8865	. . 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 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416   class class class wbr 5109  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8820  Basecbs 17264  lecple 17312  Monotcmnt 33298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8822  df-mnt 33300

This theorem is referenced by:  ismntd  33304  mntf  33305  mgcmnt1d  33317  mgcmnt2d  33318