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Theorem ismntd 32724
Description: Property of being a monotone increasing function, deduction version. (Contributed by Thierry Arnoux, 24-Apr-2024.)
Hypotheses
Ref Expression
ismntd.1 𝐴 = (Baseβ€˜π‘‰)
ismntd.2 𝐡 = (Baseβ€˜π‘Š)
ismntd.3 ≀ = (leβ€˜π‘‰)
ismntd.4 ≲ = (leβ€˜π‘Š)
ismntd.5 (πœ‘ β†’ 𝑉 ∈ 𝐢)
ismntd.6 (πœ‘ β†’ π‘Š ∈ 𝐷)
ismntd.7 (πœ‘ β†’ 𝐹 ∈ (𝑉Monotπ‘Š))
ismntd.8 (πœ‘ β†’ 𝑋 ∈ 𝐴)
ismntd.9 (πœ‘ β†’ π‘Œ ∈ 𝐴)
ismntd.10 (πœ‘ β†’ 𝑋 ≀ π‘Œ)
Assertion
Ref Expression
ismntd (πœ‘ β†’ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘Œ))

Proof of Theorem ismntd
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismntd.5 . . 3 (πœ‘ β†’ 𝑉 ∈ 𝐢)
2 ismntd.6 . . 3 (πœ‘ β†’ π‘Š ∈ 𝐷)
3 ismntd.7 . . 3 (πœ‘ β†’ 𝐹 ∈ (𝑉Monotπ‘Š))
4 ismntd.1 . . . . . 6 𝐴 = (Baseβ€˜π‘‰)
5 ismntd.2 . . . . . 6 𝐡 = (Baseβ€˜π‘Š)
6 ismntd.3 . . . . . 6 ≀ = (leβ€˜π‘‰)
7 ismntd.4 . . . . . 6 ≲ = (leβ€˜π‘Š)
84, 5, 6, 7ismnt 32723 . . . . 5 ((𝑉 ∈ 𝐢 ∧ π‘Š ∈ 𝐷) β†’ (𝐹 ∈ (𝑉Monotπ‘Š) ↔ (𝐹:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)))))
98biimp3a 1466 . . . 4 ((𝑉 ∈ 𝐢 ∧ π‘Š ∈ 𝐷 ∧ 𝐹 ∈ (𝑉Monotπ‘Š)) β†’ (𝐹:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦))))
109simprd 495 . . 3 ((𝑉 ∈ 𝐢 ∧ π‘Š ∈ 𝐷 ∧ 𝐹 ∈ (𝑉Monotπ‘Š)) β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)))
111, 2, 3, 10syl3anc 1369 . 2 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)))
12 ismntd.10 . 2 (πœ‘ β†’ 𝑋 ≀ π‘Œ)
13 breq1 5151 . . . 4 (π‘₯ = 𝑋 β†’ (π‘₯ ≀ 𝑦 ↔ 𝑋 ≀ 𝑦))
14 fveq2 6897 . . . . 5 (π‘₯ = 𝑋 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))
1514breq1d 5158 . . . 4 (π‘₯ = 𝑋 β†’ ((πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦) ↔ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘¦)))
1613, 15imbi12d 344 . . 3 (π‘₯ = 𝑋 β†’ ((π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)) ↔ (𝑋 ≀ 𝑦 β†’ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘¦))))
17 breq2 5152 . . . 4 (𝑦 = π‘Œ β†’ (𝑋 ≀ 𝑦 ↔ 𝑋 ≀ π‘Œ))
18 fveq2 6897 . . . . 5 (𝑦 = π‘Œ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘Œ))
1918breq2d 5160 . . . 4 (𝑦 = π‘Œ β†’ ((πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘¦) ↔ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘Œ)))
2017, 19imbi12d 344 . . 3 (𝑦 = π‘Œ β†’ ((𝑋 ≀ 𝑦 β†’ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘¦)) ↔ (𝑋 ≀ π‘Œ β†’ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘Œ))))
21 ismntd.8 . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐴)
22 eqidd 2729 . . 3 ((πœ‘ ∧ π‘₯ = 𝑋) β†’ 𝐴 = 𝐴)
23 ismntd.9 . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐴)
2416, 20, 21, 22, 23rspc2vd 3943 . 2 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)) β†’ (𝑋 ≀ π‘Œ β†’ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘Œ))))
2511, 12, 24mp2d 49 1 (πœ‘ β†’ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058   class class class wbr 5148  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420  Basecbs 17180  lecple 17240  Monotcmnt 32718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8847  df-mnt 32720
This theorem is referenced by:  mgcmntco  32734  mgcf1o  32743
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