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Theorem ismnt 32140
Description: Express the statement "𝐹 is monotone". (Contributed by Thierry Arnoux, 23-Apr-2024.)
Hypotheses
Ref Expression
mntoval.1 𝐴 = (Baseβ€˜π‘‰)
mntoval.2 𝐡 = (Baseβ€˜π‘Š)
mntoval.3 ≀ = (leβ€˜π‘‰)
mntoval.4 ≲ = (leβ€˜π‘Š)
Assertion
Ref Expression
ismnt ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝐹 ∈ (𝑉Monotπ‘Š) ↔ (𝐹:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)))))
Distinct variable groups:   π‘₯,𝐴,𝑦   π‘₯,𝑉,𝑦   π‘₯,π‘Š,𝑦   π‘₯,𝐹,𝑦
Allowed substitution hints:   𝐡(π‘₯,𝑦)   ≀ (π‘₯,𝑦)   𝑋(π‘₯,𝑦)   π‘Œ(π‘₯,𝑦)   ≲ (π‘₯,𝑦)

Proof of Theorem ismnt
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 mntoval.1 . . . . 5 𝐴 = (Baseβ€˜π‘‰)
2 mntoval.2 . . . . 5 𝐡 = (Baseβ€˜π‘Š)
3 mntoval.3 . . . . 5 ≀ = (leβ€˜π‘‰)
4 mntoval.4 . . . . 5 ≲ = (leβ€˜π‘Š)
51, 2, 3, 4mntoval 32139 . . . 4 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝑉Monotπ‘Š) = {𝑓 ∈ (𝐡 ↑m 𝐴) ∣ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦))})
65eleq2d 2819 . . 3 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝐹 ∈ (𝑉Monotπ‘Š) ↔ 𝐹 ∈ {𝑓 ∈ (𝐡 ↑m 𝐴) ∣ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦))}))
7 fveq1 6887 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
8 fveq1 6887 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘¦) = (πΉβ€˜π‘¦))
97, 8breq12d 5160 . . . . . 6 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦) ↔ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)))
109imbi2d 340 . . . . 5 (𝑓 = 𝐹 β†’ ((π‘₯ ≀ 𝑦 β†’ (π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦)) ↔ (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦))))
11102ralbidv 3218 . . . 4 (𝑓 = 𝐹 β†’ (βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦))))
1211elrab 3682 . . 3 (𝐹 ∈ {𝑓 ∈ (𝐡 ↑m 𝐴) ∣ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦))} ↔ (𝐹 ∈ (𝐡 ↑m 𝐴) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦))))
136, 12bitrdi 286 . 2 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝐹 ∈ (𝑉Monotπ‘Š) ↔ (𝐹 ∈ (𝐡 ↑m 𝐴) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)))))
142fvexi 6902 . . . 4 𝐡 ∈ V
151fvexi 6902 . . . 4 𝐴 ∈ V
1614, 15elmap 8861 . . 3 (𝐹 ∈ (𝐡 ↑m 𝐴) ↔ 𝐹:𝐴⟢𝐡)
1716anbi1i 624 . 2 ((𝐹 ∈ (𝐡 ↑m 𝐴) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦))) ↔ (𝐹:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦))))
1813, 17bitrdi 286 1 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝐹 ∈ (𝑉Monotπ‘Š) ↔ (𝐹:𝐴⟢𝐡 ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (πΉβ€˜π‘₯) ≲ (πΉβ€˜π‘¦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432   class class class wbr 5147  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405   ↑m cmap 8816  Basecbs 17140  lecple 17200  Monotcmnt 32135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-mnt 32137
This theorem is referenced by:  ismntd  32141  mntf  32142  mgcmnt1d  32154  mgcmnt2d  32155
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