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Theorem ismntd 30699
 Description: Property of being a monotone increasing function, deduction version. (Contributed by Thierry Arnoux, 24-Apr-2024.)
Hypotheses
Ref Expression
ismntd.1 𝐴 = (Base‘𝑉)
ismntd.2 𝐵 = (Base‘𝑊)
ismntd.3 = (le‘𝑉)
ismntd.4 = (le‘𝑊)
ismntd.5 (𝜑𝑉𝐶)
ismntd.6 (𝜑𝑊𝐷)
ismntd.7 (𝜑𝐹 ∈ (𝑉Monot𝑊))
ismntd.8 (𝜑𝑋𝐴)
ismntd.9 (𝜑𝑌𝐴)
ismntd.10 (𝜑𝑋 𝑌)
Assertion
Ref Expression
ismntd (𝜑 → (𝐹𝑋) (𝐹𝑌))

Proof of Theorem ismntd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismntd.5 . . 3 (𝜑𝑉𝐶)
2 ismntd.6 . . 3 (𝜑𝑊𝐷)
3 ismntd.7 . . 3 (𝜑𝐹 ∈ (𝑉Monot𝑊))
4 ismntd.1 . . . . . 6 𝐴 = (Base‘𝑉)
5 ismntd.2 . . . . . 6 𝐵 = (Base‘𝑊)
6 ismntd.3 . . . . . 6 = (le‘𝑉)
7 ismntd.4 . . . . . 6 = (le‘𝑊)
84, 5, 6, 7ismnt 30698 . . . . 5 ((𝑉𝐶𝑊𝐷) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))))
98biimp3a 1466 . . . 4 ((𝑉𝐶𝑊𝐷𝐹 ∈ (𝑉Monot𝑊)) → (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
109simprd 499 . . 3 ((𝑉𝐶𝑊𝐷𝐹 ∈ (𝑉Monot𝑊)) → ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))
111, 2, 3, 10syl3anc 1368 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))
12 ismntd.10 . 2 (𝜑𝑋 𝑌)
13 breq1 5033 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
14 fveq2 6645 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1514breq1d 5040 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑦)))
1613, 15imbi12d 348 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)) ↔ (𝑋 𝑦 → (𝐹𝑋) (𝐹𝑦))))
17 breq2 5034 . . . 4 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
18 fveq2 6645 . . . . 5 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1918breq2d 5042 . . . 4 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑌)))
2017, 19imbi12d 348 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑦 → (𝐹𝑋) (𝐹𝑦)) ↔ (𝑋 𝑌 → (𝐹𝑋) (𝐹𝑌))))
21 ismntd.8 . . 3 (𝜑𝑋𝐴)
22 eqidd 2799 . . 3 ((𝜑𝑥 = 𝑋) → 𝐴 = 𝐴)
23 ismntd.9 . . 3 (𝜑𝑌𝐴)
2416, 20, 21, 22, 23rspc2vd 3877 . 2 (𝜑 → (∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)) → (𝑋 𝑌 → (𝐹𝑋) (𝐹𝑌))))
2511, 12, 24mp2d 49 1 (𝜑 → (𝐹𝑋) (𝐹𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   class class class wbr 5030  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  Basecbs 16477  lecple 16566  Monotcmnt 30693 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 7443 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 8393  df-mnt 30695 This theorem is referenced by:  mgcmntco  30709
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