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Theorem ismntd 31893
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 31892 . . . . 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 1470 . . . 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 497 . . 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 1372 . 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 5109 . . . 4 (π‘₯ = 𝑋 β†’ (π‘₯ ≀ 𝑦 ↔ 𝑋 ≀ 𝑦))
14 fveq2 6843 . . . . 5 (π‘₯ = 𝑋 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))
1514breq1d 5116 . . . 4 (π‘₯ = 𝑋 β†’ ((πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦) ↔ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘¦)))
1613, 15imbi12d 345 . . 3 (π‘₯ = 𝑋 β†’ ((π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)) ↔ (𝑋 ≀ 𝑦 β†’ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘¦))))
17 breq2 5110 . . . 4 (𝑦 = π‘Œ β†’ (𝑋 ≀ 𝑦 ↔ 𝑋 ≀ π‘Œ))
18 fveq2 6843 . . . . 5 (𝑦 = π‘Œ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘Œ))
1918breq2d 5118 . . . 4 (𝑦 = π‘Œ β†’ ((πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘¦) ↔ (πΉβ€˜π‘‹) ≲ (πΉβ€˜π‘Œ)))
2017, 19imbi12d 345 . . 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 2734 . . 3 ((πœ‘ ∧ π‘₯ = 𝑋) β†’ 𝐴 = 𝐴)
23 ismntd.9 . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐴)
2416, 20, 21, 22, 23rspc2vd 3907 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   class class class wbr 5106  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145  Monotcmnt 31887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-mnt 31889
This theorem is referenced by:  mgcmntco  31903  mgcf1o  31912
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