Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ismntd Structured version   Visualization version   GIF version

Theorem ismntd 32657
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 32656 . . . . 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 1465 . . . 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 1368 . 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 5144 . . . 4 (π‘₯ = 𝑋 β†’ (π‘₯ ≀ 𝑦 ↔ 𝑋 ≀ 𝑦))
14 fveq2 6884 . . . . 5 (π‘₯ = 𝑋 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘‹))
1514breq1d 5151 . . . 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 5145 . . . 4 (𝑦 = π‘Œ β†’ (𝑋 ≀ 𝑦 ↔ 𝑋 ≀ π‘Œ))
18 fveq2 6884 . . . . 5 (𝑦 = π‘Œ β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘Œ))
1918breq2d 5153 . . . 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 2727 . . 3 ((πœ‘ ∧ π‘₯ = 𝑋) β†’ 𝐴 = 𝐴)
23 ismntd.9 . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐴)
2416, 20, 21, 22, 23rspc2vd 3939 . 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 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055   class class class wbr 5141  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404  Basecbs 17151  lecple 17211  Monotcmnt 32651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-mnt 32653
This theorem is referenced by:  mgcmntco  32667  mgcf1o  32676
  Copyright terms: Public domain W3C validator