Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ismnt | Structured version Visualization version GIF version |
Description: Express the statement "𝐹 is monotone " (Contributed by Thierry Arnoux, 23-Apr-2024.) |
Ref | Expression |
---|---|
mntoval.1 | ⊢ 𝐴 = (Base‘𝑉) |
mntoval.2 | ⊢ 𝐵 = (Base‘𝑊) |
mntoval.3 | ⊢ ≤ = (le‘𝑉) |
mntoval.4 | ⊢ ≲ = (le‘𝑊) |
Ref | Expression |
---|---|
ismnt | ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mntoval.1 | . . . . 5 ⊢ 𝐴 = (Base‘𝑉) | |
2 | mntoval.2 | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
3 | mntoval.3 | . . . . 5 ⊢ ≤ = (le‘𝑉) | |
4 | mntoval.4 | . . . . 5 ⊢ ≲ = (le‘𝑊) | |
5 | 1, 2, 3, 4 | mntoval 30651 | . . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))}) |
6 | 5 | eleq2d 2896 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ 𝐹 ∈ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))})) |
7 | fveq1 6645 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) | |
8 | fveq1 6645 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦)) | |
9 | 7, 8 | breq12d 5055 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ≲ (𝑓‘𝑦) ↔ (𝐹‘𝑥) ≲ (𝐹‘𝑦))) |
10 | 9 | imbi2d 343 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦)) ↔ (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))) |
11 | 10 | 2ralbidv 3186 | . . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))) |
12 | 11 | elrab 3660 | . . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝑓‘𝑥) ≲ (𝑓‘𝑦))} ↔ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))) |
13 | 6, 12 | syl6bb 289 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹 ∈ (𝐵 ↑m 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))))) |
14 | 2 | fvexi 6660 | . . . 4 ⊢ 𝐵 ∈ V |
15 | 1 | fvexi 6660 | . . . 4 ⊢ 𝐴 ∈ V |
16 | 14, 15 | elmap 8413 | . . 3 ⊢ (𝐹 ∈ (𝐵 ↑m 𝐴) ↔ 𝐹:𝐴⟶𝐵) |
17 | 16 | anbi1i 625 | . 2 ⊢ ((𝐹 ∈ (𝐵 ↑m 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦)))) |
18 | 13, 17 | syl6bb 289 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≤ 𝑦 → (𝐹‘𝑥) ≲ (𝐹‘𝑦))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3125 {crab 3129 class class class wbr 5042 ⟶wf 6327 ‘cfv 6331 (class class class)co 7133 ↑m cmap 8384 Basecbs 16462 lecple 16551 Monotcmnt 30647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5179 ax-nul 5186 ax-pow 5242 ax-pr 5306 ax-un 7439 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ral 3130 df-rex 3131 df-rab 3134 df-v 3475 df-sbc 3753 df-csb 3861 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4270 df-if 4444 df-pw 4517 df-sn 4544 df-pr 4546 df-op 4550 df-uni 4815 df-br 5043 df-opab 5105 df-id 5436 df-xp 5537 df-rel 5538 df-cnv 5539 df-co 5540 df-dm 5541 df-rn 5542 df-iota 6290 df-fun 6333 df-fn 6334 df-f 6335 df-fv 6339 df-ov 7136 df-oprab 7137 df-mpo 7138 df-map 8386 df-mnt 30649 |
This theorem is referenced by: ismntd 30653 mntf 30654 |
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