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Theorem ismnt 32957
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 32956 . . . 4 ((𝑉𝑋𝑊𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))})
65eleq2d 2824 . . 3 ((𝑉𝑋𝑊𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ 𝐹 ∈ {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))}))
7 fveq1 6905 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
8 fveq1 6905 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
97, 8breq12d 5160 . . . . . 6 (𝑓 = 𝐹 → ((𝑓𝑥) (𝑓𝑦) ↔ (𝐹𝑥) (𝐹𝑦)))
109imbi2d 340 . . . . 5 (𝑓 = 𝐹 → ((𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦)) ↔ (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
11102ralbidv 3218 . . . 4 (𝑓 = 𝐹 → (∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
1211elrab 3694 . . 3 (𝐹 ∈ {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))} ↔ (𝐹 ∈ (𝐵m 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
136, 12bitrdi 287 . 2 ((𝑉𝑋𝑊𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹 ∈ (𝐵m 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))))
142fvexi 6920 . . . 4 𝐵 ∈ V
151fvexi 6920 . . . 4 𝐴 ∈ V
1614, 15elmap 8909 . . 3 (𝐹 ∈ (𝐵m 𝐴) ↔ 𝐹:𝐴𝐵)
1716anbi1i 624 . 2 ((𝐹 ∈ (𝐵m 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
1813, 17bitrdi 287 1 ((𝑉𝑋𝑊𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  {crab 3432   class class class wbr 5147  wf 6558  cfv 6562  (class class class)co 7430  m cmap 8864  Basecbs 17244  lecple 17304  Monotcmnt 32952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-mnt 32954
This theorem is referenced by:  ismntd  32958  mntf  32959  mgcmnt1d  32971  mgcmnt2d  32972
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