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Theorem ismntd 30981
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 30980 . . . . 5 ((𝑉𝐶𝑊𝐷) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))))
98biimp3a 1471 . . . 4 ((𝑉𝐶𝑊𝐷𝐹 ∈ (𝑉Monot𝑊)) → (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
109simprd 499 . . 3 ((𝑉𝐶𝑊𝐷𝐹 ∈ (𝑉Monot𝑊)) → ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))
111, 2, 3, 10syl3anc 1373 . 2 (𝜑 → ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))
12 ismntd.10 . 2 (𝜑𝑋 𝑌)
13 breq1 5056 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
14 fveq2 6717 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1514breq1d 5063 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑦)))
1613, 15imbi12d 348 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)) ↔ (𝑋 𝑦 → (𝐹𝑋) (𝐹𝑦))))
17 breq2 5057 . . . 4 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
18 fveq2 6717 . . . . 5 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1918breq2d 5065 . . . 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 2738 . . 3 ((𝜑𝑥 = 𝑋) → 𝐴 = 𝐴)
23 ismntd.9 . . 3 (𝜑𝑌𝐴)
2416, 20, 21, 22, 23rspc2vd 3862 . 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 1089   = wceq 1543  wcel 2110  wral 3061   class class class wbr 5053  wf 6376  cfv 6380  (class class class)co 7213  Basecbs 16760  lecple 16809  Monotcmnt 30975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-map 8510  df-mnt 30977
This theorem is referenced by:  mgcmntco  30991  mgcf1o  31000
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