Theorem ismntd 33070

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 33069	. . . . 5 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))))
9	8	biimp3a 1477	. . . 4 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝐹 ∈ (𝑉Monot𝑊)) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))))
10	9	simprd 496	. . 3 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝐹 ∈ (𝑉Monot𝑊)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
11	1, 2, 3, 10	syl3anc 1379	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
12		ismntd.10	. 2 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
13		breq1 5082	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
14		fveq2 6834	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
15	14	breq1d 5089	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑦)))
16	13, 15	imbi12d 345	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑦 → (𝐹‘𝑋) ≲ (𝐹‘𝑦))))
17		breq2 5083	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
18		fveq2 6834	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
19	18	breq2d 5091	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑌)))
20	17, 19	imbi12d 345	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 → (𝐹‘𝑋) ≲ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑌 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))))
21		ismntd.8	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
22		eqidd 2741	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐴 = 𝐴)
23		ismntd.9	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
24	16, 20, 21, 22, 23	rspc2vd 3886	. 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 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054   class class class wbr 5079  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  lecple 17225  Monotcmnt 33064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-map 8772  df-mnt 33066

This theorem is referenced by:  mgcmntco  33080  mgcf1o  33089