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Theorem mntf 32625
Description: A monotone function is a function. (Contributed by Thierry Arnoux, 24-Apr-2024.)
Hypotheses
Ref Expression
mntf.1 𝐴 = (Baseβ€˜π‘‰)
mntf.2 𝐡 = (Baseβ€˜π‘Š)
Assertion
Ref Expression
mntf ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ ∧ 𝐹 ∈ (𝑉Monotπ‘Š)) β†’ 𝐹:𝐴⟢𝐡)

Proof of Theorem mntf
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mntf.1 . . . 4 𝐴 = (Baseβ€˜π‘‰)
2 mntf.2 . . . 4 𝐡 = (Baseβ€˜π‘Š)
3 eqid 2724 . . . 4 (leβ€˜π‘‰) = (leβ€˜π‘‰)
4 eqid 2724 . . . 4 (leβ€˜π‘Š) = (leβ€˜π‘Š)
51, 2, 3, 4ismnt 32623 . . 3 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝐹 ∈ (𝑉Monotπ‘Š) ↔ (𝐹:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯(leβ€˜π‘‰)𝑦 β†’ (πΉβ€˜π‘₯)(leβ€˜π‘Š)(πΉβ€˜π‘¦)))))
65biimp3a 1465 . 2 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ ∧ 𝐹 ∈ (𝑉Monotπ‘Š)) β†’ (𝐹:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯(leβ€˜π‘‰)𝑦 β†’ (πΉβ€˜π‘₯)(leβ€˜π‘Š)(πΉβ€˜π‘¦))))
76simpld 494 1 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ ∧ 𝐹 ∈ (𝑉Monotπ‘Š)) β†’ 𝐹:𝐴⟢𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053   class class class wbr 5139  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402  Basecbs 17145  lecple 17205  Monotcmnt 32618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-map 8819  df-mnt 32620
This theorem is referenced by:  mgcmntco  32634
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