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Theorem ismnt 33316
Description: Express the statement "𝐹 is monotone". (Contributed by Thierry Arnoux, 23-Apr-2024.)
Hypotheses
Ref Expression
mntoval.1 𝐴 = (Base‘𝑉)
mntoval.2 𝐵 = (Base‘𝑊)
mntoval.3 = (le‘𝑉)
mntoval.4 = (le‘𝑊)
Assertion
Ref Expression
ismnt ((𝑉𝑋𝑊𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐵(𝑥, 𝑦)   (𝑥, 𝑦)   𝑋(𝑥, 𝑦)   𝑌(𝑥, 𝑦)   (𝑥, 𝑦)

Proof of Theorem ismnt
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef 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)
51, 2, 3, 4mntoval 33315 . . . 4 ((𝑉𝑋𝑊𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))})
65eleq2d 2852 . . 3 ((𝑉𝑋𝑊𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ 𝐹 ∈ {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))}))
7 fveq1 6884 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
8 fveq1 6884 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
97, 8breq12d 5125 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑥) (𝑓𝑦) ↔ (𝐹𝑥) (𝐹𝑦)))
109imbi2d 343 . . . . 5 (𝑓 = 𝐹 → ((𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦)) ↔ (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
11102ralbidv 3232 . . . 4 (𝑓 = 𝐹 → (∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
1211elrab 3653 . . 3 (𝐹 ∈ {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))} ↔ (𝐹 ∈ (𝐵m 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
136, 12bitrdi 290 . 2 ((𝑉𝑋𝑊𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹 ∈ (𝐵m 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))))
142fvexi 6899 . . . 4 𝐵 ∈ V
151fvexi 6899 . . . 4 𝐴 ∈ V
1614, 15elmap 8871 . . 3 (𝐹 ∈ (𝐵m 𝐴) ↔ 𝐹:𝐴𝐵)
1716anbi1i 636 . 2 ((𝐹 ∈ (𝐵m 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
1813, 17bitrdi 290 1 ((𝑉𝑋𝑊𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))))
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wi 4  wb 209  wa 401   = wceq 1570  wcel 2146  wral 3082  {crab 3419   class class class wbr 5112  wf 6536  cfv 6540  (class class class)co 7416  m cmap 8826  Basecbs 17279  lecple 17327  Monotcmnt 33311
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5260  ax-nul 5272  ax-pow 5339  ax-pr 5407  ax-un 7738
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4876  df-br 5113  df-opab 5177  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-map 8828  df-mnt 33313
This theorem is used by:  ismntd  33317  mntf  33318  mgcmnt1d  33330  mgcmnt2d  33331
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