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Theorem mntf 32959
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 2734 . . . 4 (le‘𝑉) = (le‘𝑉)
4 eqid 2734 . . . 4 (le‘𝑊) = (le‘𝑊)
51, 2, 3, 4ismnt 32957 . . 3 ((𝑉𝑋𝑊𝑌) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥(le‘𝑉)𝑦 → (𝐹𝑥)(le‘𝑊)(𝐹𝑦)))))
65biimp3a 1468 . 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 1086   = wceq 1536  wcel 2105  wral 3058   class class class wbr 5147  wf 6558  cfv 6562  (class class class)co 7430  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:  mgcmntco  32968
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