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Theorem ismnt 30691
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 30690 . . . 4 ((𝑉𝑋𝑊𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))})
65eleq2d 2875 . . 3 ((𝑉𝑋𝑊𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ 𝐹 ∈ {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))}))
7 fveq1 6644 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
8 fveq1 6644 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
97, 8breq12d 5043 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑥) (𝑓𝑦) ↔ (𝐹𝑥) (𝐹𝑦)))
109imbi2d 344 . . . . 5 (𝑓 = 𝐹 → ((𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦)) ↔ (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
11102ralbidv 3164 . . . 4 (𝑓 = 𝐹 → (∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
1211elrab 3628 . . 3 (𝐹 ∈ {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))} ↔ (𝐹 ∈ (𝐵m 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
136, 12syl6bb 290 . 2 ((𝑉𝑋𝑊𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹 ∈ (𝐵m 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))))
142fvexi 6659 . . . 4 𝐵 ∈ V
151fvexi 6659 . . . 4 𝐴 ∈ V
1614, 15elmap 8418 . . 3 (𝐹 ∈ (𝐵m 𝐴) ↔ 𝐹:𝐴𝐵)
1716anbi1i 626 . 2 ((𝐹 ∈ (𝐵m 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
1813, 17syl6bb 290 1 ((𝑉𝑋𝑊𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wral 3106  {crab 3110   class class class wbr 5030  wf 6320  cfv 6324  (class class class)co 7135  m cmap 8389  Basecbs 16475  lecple 16564  Monotcmnt 30686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-mnt 30688
This theorem is referenced by:  ismntd  30692  mntf  30693
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