Theorem ismntd 32935

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 32934	. . . . 5 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))))
9	8	biimp3a 1471	. . . 4 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝐹 ∈ (𝑉Monot𝑊)) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))))
10	9	simprd 495	. . 3 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝐹 ∈ (𝑉Monot𝑊)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
11	1, 2, 3, 10	syl3anc 1373	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
12		ismntd.10	. 2 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
13		breq1 5095	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
14		fveq2 6822	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
15	14	breq1d 5102	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑦)))
16	13, 15	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑦 → (𝐹‘𝑋) ≲ (𝐹‘𝑦))))
17		breq2 5096	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
18		fveq2 6822	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
19	18	breq2d 5104	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑌)))
20	17, 19	imbi12d 344	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 → (𝐹‘𝑋) ≲ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑌 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))))
21		ismntd.8	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
22		eqidd 2730	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐴 = 𝐴)
23		ismntd.9	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
24	16, 20, 21, 22, 23	rspc2vd 3899	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) → (𝑋 ≤ 𝑌 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))))
25	11, 12, 24	mp2d 49	1 ⊢ (𝜑 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044   class class class wbr 5092  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  lecple 17168  Monotcmnt 32929

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-pow 5304  ax-pr 5371  ax-un 7671

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-rn 5630  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-map 8755  df-mnt 32931

This theorem is referenced by:  mgcmntco  32945  mgcf1o  32954