Theorem ismntd 33344

Description: Property of being a monotone increasing function, deduction version. (Contributed by Thierry Arnoux, 24-Apr-2024.)

Hypotheses

Ref	Expression
ismntd.1	⊢ 𝐴 = (Base‘𝑉)
ismntd.2	⊢ 𝐵 = (Base‘𝑊)
ismntd.3	⊢ ≤ = (le‘𝑉)
ismntd.4	⊢ ≲ = (le‘𝑊)
ismntd.5	⊢ (𝜑 → 𝑉 ∈ 𝐶)
ismntd.6	⊢ (𝜑 → 𝑊 ∈ 𝐷)
ismntd.7	⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊))
ismntd.8	⊢ (𝜑 → 𝑋 ∈ 𝐴)
ismntd.9	⊢ (𝜑 → 𝑌 ∈ 𝐴)
ismntd.10	⊢ (𝜑 → 𝑋 ≤ 𝑌)

Assertion

Ref	Expression
ismntd	⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))

Proof of Theorem ismntd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismntd.5	. . 3 ⊢ (𝜑 → 𝑉 ∈ 𝐶)
2		ismntd.6	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝐷)
3		ismntd.7	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑉Monot𝑊))
4		ismntd.1	. . . . . 6 ⊢ 𝐴 = (Base‘𝑉)
5		ismntd.2	. . . . . 6 ⊢ 𝐵 = (Base‘𝑊)
6		ismntd.3	. . . . . 6 ⊢ ≤ = (le‘𝑉)
7		ismntd.4	. . . . . 6 ⊢ ≲ = (le‘𝑊)
8	4, 5, 6, 7	ismnt 33343	. . . . 5 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))))
9	8	biimp3a 1498	. . . 4 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝐹 ∈ (𝑉Monot𝑊)) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))))
10	9	simprd 501	. . 3 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝐹 ∈ (𝑉Monot𝑊)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
11	1, 2, 3, 10	syl3anc 1398	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
12		ismntd.10	. 2 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
13		breq1 5114	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
14		fveq2 6885	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
15	14	breq1d 5121	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑦)))
16	13, 15	imbi12d 347	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑦 → (𝐹‘𝑋) ≲ (𝐹‘𝑦))))
17		breq2 5115	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
18		fveq2 6885	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
19	18	breq2d 5123	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑌)))
20	17, 19	imbi12d 347	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 → (𝐹‘𝑋) ≲ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑌 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))))
21		ismntd.8	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
22		eqidd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐴 = 𝐴)
23		ismntd.9	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
24	16, 20, 21, 22, 23	rspc2vd 3902	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) → (𝑋 ≤ 𝑌 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))))
25	11, 12, 24	mp2d 50	1 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081   class class class wbr 5111  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416  Basecbs 17287  lecple 17335  Monotcmnt 33338

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8828  df-mnt 33340

This theorem is used by:  mgcmntco  33354  mgcf1o  33363