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Theorem mntoval 32730
Description: Operation value of the monotone function. (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
mntoval ((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‘¦))})
Distinct variable groups:   π‘₯,𝐴,𝑦,𝑓   𝐡,𝑓   𝑓,𝑉,π‘₯,𝑦   π‘₯,π‘Š,𝑦,𝑓
Allowed substitution hints:   𝐡(π‘₯,𝑦)   ≀ (π‘₯,𝑦,𝑓)   𝑋(π‘₯,𝑦,𝑓)   π‘Œ(π‘₯,𝑦,𝑓)   ≲ (π‘₯,𝑦,𝑓)

Proof of Theorem mntoval
Dummy variables π‘Ž 𝑣 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mnt 32728 . . 3 Monot = (𝑣 ∈ V, 𝑀 ∈ V ↦ ⦋(Baseβ€˜π‘£) / π‘Žβ¦Œ{𝑓 ∈ ((Baseβ€˜π‘€) ↑m π‘Ž) ∣ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ π‘Ž (π‘₯(leβ€˜π‘£)𝑦 β†’ (π‘“β€˜π‘₯)(leβ€˜π‘€)(π‘“β€˜π‘¦))})
21a1i 11 . 2 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ Monot = (𝑣 ∈ V, 𝑀 ∈ V ↦ ⦋(Baseβ€˜π‘£) / π‘Žβ¦Œ{𝑓 ∈ ((Baseβ€˜π‘€) ↑m π‘Ž) ∣ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ π‘Ž (π‘₯(leβ€˜π‘£)𝑦 β†’ (π‘“β€˜π‘₯)(leβ€˜π‘€)(π‘“β€˜π‘¦))}))
3 fvexd 6917 . . . 4 ((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) β†’ (Baseβ€˜π‘£) ∈ V)
4 fveq2 6902 . . . . . 6 (𝑣 = 𝑉 β†’ (Baseβ€˜π‘£) = (Baseβ€˜π‘‰))
5 mntoval.1 . . . . . 6 𝐴 = (Baseβ€˜π‘‰)
64, 5eqtr4di 2786 . . . . 5 (𝑣 = 𝑉 β†’ (Baseβ€˜π‘£) = 𝐴)
76adantr 479 . . . 4 ((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) β†’ (Baseβ€˜π‘£) = 𝐴)
8 simplr 767 . . . . . . . 8 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ 𝑀 = π‘Š)
98fveq2d 6906 . . . . . . 7 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
10 mntoval.2 . . . . . . 7 𝐡 = (Baseβ€˜π‘Š)
119, 10eqtr4di 2786 . . . . . 6 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (Baseβ€˜π‘€) = 𝐡)
12 simpr 483 . . . . . 6 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ π‘Ž = 𝐴)
1311, 12oveq12d 7444 . . . . 5 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ ((Baseβ€˜π‘€) ↑m π‘Ž) = (𝐡 ↑m 𝐴))
14 simpll 765 . . . . . . . . . . 11 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ 𝑣 = 𝑉)
1514fveq2d 6906 . . . . . . . . . 10 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (leβ€˜π‘£) = (leβ€˜π‘‰))
16 mntoval.3 . . . . . . . . . 10 ≀ = (leβ€˜π‘‰)
1715, 16eqtr4di 2786 . . . . . . . . 9 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (leβ€˜π‘£) = ≀ )
1817breqd 5163 . . . . . . . 8 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (π‘₯(leβ€˜π‘£)𝑦 ↔ π‘₯ ≀ 𝑦))
198fveq2d 6906 . . . . . . . . . 10 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (leβ€˜π‘€) = (leβ€˜π‘Š))
20 mntoval.4 . . . . . . . . . 10 ≲ = (leβ€˜π‘Š)
2119, 20eqtr4di 2786 . . . . . . . . 9 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (leβ€˜π‘€) = ≲ )
2221breqd 5163 . . . . . . . 8 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ ((π‘“β€˜π‘₯)(leβ€˜π‘€)(π‘“β€˜π‘¦) ↔ (π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦)))
2318, 22imbi12d 343 . . . . . . 7 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ ((π‘₯(leβ€˜π‘£)𝑦 β†’ (π‘“β€˜π‘₯)(leβ€˜π‘€)(π‘“β€˜π‘¦)) ↔ (π‘₯ ≀ 𝑦 β†’ (π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦))))
2412, 23raleqbidv 3340 . . . . . 6 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (βˆ€π‘¦ ∈ π‘Ž (π‘₯(leβ€˜π‘£)𝑦 β†’ (π‘“β€˜π‘₯)(leβ€˜π‘€)(π‘“β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦))))
2512, 24raleqbidv 3340 . . . . 5 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ π‘Ž (π‘₯(leβ€˜π‘£)𝑦 β†’ (π‘“β€˜π‘₯)(leβ€˜π‘€)(π‘“β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦))))
2613, 25rabeqbidv 3448 . . . 4 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ {𝑓 ∈ ((Baseβ€˜π‘€) ↑m π‘Ž) ∣ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ π‘Ž (π‘₯(leβ€˜π‘£)𝑦 β†’ (π‘“β€˜π‘₯)(leβ€˜π‘€)(π‘“β€˜π‘¦))} = {𝑓 ∈ (𝐡 ↑m 𝐴) ∣ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦))})
273, 7, 26csbied2 3934 . . 3 ((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) β†’ ⦋(Baseβ€˜π‘£) / π‘Žβ¦Œ{𝑓 ∈ ((Baseβ€˜π‘€) ↑m π‘Ž) ∣ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ π‘Ž (π‘₯(leβ€˜π‘£)𝑦 β†’ (π‘“β€˜π‘₯)(leβ€˜π‘€)(π‘“β€˜π‘¦))} = {𝑓 ∈ (𝐡 ↑m 𝐴) ∣ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦))})
2827adantl 480 . 2 (((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) ∧ (𝑣 = 𝑉 ∧ 𝑀 = π‘Š)) β†’ ⦋(Baseβ€˜π‘£) / π‘Žβ¦Œ{𝑓 ∈ ((Baseβ€˜π‘€) ↑m π‘Ž) ∣ βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ π‘Ž (π‘₯(leβ€˜π‘£)𝑦 β†’ (π‘“β€˜π‘₯)(leβ€˜π‘€)(π‘“β€˜π‘¦))} = {𝑓 ∈ (𝐡 ↑m 𝐴) ∣ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦))})
29 elex 3492 . . 3 (𝑉 ∈ 𝑋 β†’ 𝑉 ∈ V)
3029adantr 479 . 2 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ 𝑉 ∈ V)
31 elex 3492 . . 3 (π‘Š ∈ π‘Œ β†’ π‘Š ∈ V)
3231adantl 480 . 2 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ π‘Š ∈ V)
33 eqid 2728 . . 3 {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‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦))}
34 ovexd 7461 . . 3 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝐡 ↑m 𝐴) ∈ V)
3533, 34rabexd 5339 . 2 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ {𝑓 ∈ (𝐡 ↑m 𝐴) ∣ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐴 (π‘₯ ≀ 𝑦 β†’ (π‘“β€˜π‘₯) ≲ (π‘“β€˜π‘¦))} ∈ V)
362, 28, 30, 32, 35ovmpod 7579 1 ((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‘¦))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {crab 3430  Vcvv 3473  β¦‹csb 3894   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426   ∈ cmpo 7428   ↑m cmap 8851  Basecbs 17187  lecple 17247  Monotcmnt 32726
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-mnt 32728
This theorem is referenced by:  ismnt  32731
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