Theorem ismntd 33123

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 33122	. . . . 5 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))))
9	8	biimp3a 1489	. . . 4 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝐹 ∈ (𝑉Monot𝑊)) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))))
10	9	simprd 499	. . 3 ⊢ ((𝑉 ∈ 𝐶 ∧ 𝑊 ∈ 𝐷 ∧ 𝐹 ∈ (𝑉Monot𝑊)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
11	1, 2, 3, 10	syl3anc 1389	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))
12		ismntd.10	. 2 ⊢ (𝜑 → 𝑋 ≤ 𝑌)
13		breq1 5102	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦))
14		fveq2 6863	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
15	14	breq1d 5109	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑦)))
16	13, 15	imbi12d 346	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑦 → (𝐹‘𝑋) ≲ (𝐹‘𝑦))))
17		breq2 5103	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌))
18		fveq2 6863	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
19	18	breq2d 5111	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) ≲ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≲ (𝐹‘𝑌)))
20	17, 19	imbi12d 346	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 → (𝐹‘𝑋) ≲ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑌 → (𝐹‘𝑋) ≲ (𝐹‘𝑌))))
21		ismntd.8	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
22		eqidd 2762	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐴 = 𝐴)
23		ismntd.9	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
24	16, 20, 21, 22, 23	rspc2vd 3900	. 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 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075   class class class wbr 5099  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  lecple 17276  Monotcmnt 33117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-map 8805  df-mnt 33119

This theorem is referenced by:  mgcmntco  33133  mgcf1o  33142