Theorem ismntd 32957

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 32956	. . . . 5 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))))
9	8	biimp3a 1469	. . . 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 1371	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
12		ismntd.10	. 2 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
13		breq1 5169	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
14		fveq2 6920	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
15	14	breq1d 5176	. . . 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 5170	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
18		fveq2 6920	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
19	18	breq2d 5178	. . . 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 2741	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐴 = 𝐴)
23		ismntd.9	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
24	16, 20, 21, 22, 23	rspc2vd 3972	. 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 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067   class class class wbr 5166  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  lecple 17318  Monotcmnt 32951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-mnt 32953

This theorem is referenced by:  mgcmntco  32967  mgcf1o  32976