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Theorem mntf 32975
Description: A monotone function is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
Hypotheses
Ref Expression
mntf.1 𝐴 = (Base‘𝑉)
mntf.2 𝐵 = (Base‘𝑊)
Assertion
Ref Expression
mntf ((𝑉𝑋𝑊𝑌𝐹 ∈ (𝑉Monot𝑊)) → 𝐹:𝐴𝐵)

Proof of Theorem mntf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mntf.1 . . . 4 𝐴 = (Base‘𝑉)
2 mntf.2 . . . 4 𝐵 = (Base‘𝑊)
3 eqid 2737 . . . 4 (le‘𝑉) = (le‘𝑉)
4 eqid 2737 . . . 4 (le‘𝑊) = (le‘𝑊)
51, 2, 3, 4ismnt 32973 . . 3 ((𝑉𝑋𝑊𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥(le‘𝑉)𝑦 → (𝐹𝑥)(le‘𝑊)(𝐹𝑦)))))
65biimp3a 1471 . 2 ((𝑉𝑋𝑊𝑌𝐹 ∈ (𝑉Monot𝑊)) → (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥(le‘𝑉)𝑦 → (𝐹𝑥)(le‘𝑊)(𝐹𝑦))))
76simpld 494 1 ((𝑉𝑋𝑊𝑌𝐹 ∈ (𝑉Monot𝑊)) → 𝐹:𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061   class class class wbr 5143  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  Monotcmnt 32968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-mnt 32970
This theorem is referenced by:  mgcmntco  32984
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