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Theorem ismntd 32916
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 32915 . . . . 5 ((𝑉𝐶𝑊𝐷) → (𝐹 ∈ (𝑉Monot𝑊) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)))))
98biimp3a 1471 . . . 4 ((𝑉𝐶𝑊𝐷𝐹 ∈ (𝑉Monot𝑊)) → (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦))))
109simprd 495 . . 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 5112 . . . 4 (𝑥 = 𝑋 → (𝑥 𝑦𝑋 𝑦))
14 fveq2 6860 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1514breq1d 5119 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑦)))
1613, 15imbi12d 344 . . 3 (𝑥 = 𝑋 → ((𝑥 𝑦 → (𝐹𝑥) (𝐹𝑦)) ↔ (𝑋 𝑦 → (𝐹𝑋) (𝐹𝑦))))
17 breq2 5113 . . . 4 (𝑦 = 𝑌 → (𝑋 𝑦𝑋 𝑌))
18 fveq2 6860 . . . . 5 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1918breq2d 5121 . . . 4 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦) ↔ (𝐹𝑋) (𝐹𝑌)))
2017, 19imbi12d 344 . . 3 (𝑦 = 𝑌 → ((𝑋 𝑦 → (𝐹𝑋) (𝐹𝑦)) ↔ (𝑋 𝑌 → (𝐹𝑋) (𝐹𝑌))))
21 ismntd.8 . . 3 (𝜑𝑋𝐴)
22 eqidd 2731 . . 3 ((𝜑𝑥 = 𝑋) → 𝐴 = 𝐴)
23 ismntd.9 . . 3 (𝜑𝑌𝐴)
2416, 20, 21, 22, 23rspc2vd 3912 . 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 395  w3a 1086   = wceq 1540  wcel 2109  wral 3045   class class class wbr 5109  wf 6509  cfv 6513  (class class class)co 7389  Basecbs 17185  lecple 17233  Monotcmnt 32910
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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394  df-map 8803  df-mnt 32912
This theorem is referenced by:  mgcmntco  32926  mgcf1o  32935
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