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Theorem mntoval 32654
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 32652 . . 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 6899 . . . 4 ((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) β†’ (Baseβ€˜π‘£) ∈ V)
4 fveq2 6884 . . . . . 6 (𝑣 = 𝑉 β†’ (Baseβ€˜π‘£) = (Baseβ€˜π‘‰))
5 mntoval.1 . . . . . 6 𝐴 = (Baseβ€˜π‘‰)
64, 5eqtr4di 2784 . . . . 5 (𝑣 = 𝑉 β†’ (Baseβ€˜π‘£) = 𝐴)
76adantr 480 . . . 4 ((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) β†’ (Baseβ€˜π‘£) = 𝐴)
8 simplr 766 . . . . . . . 8 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ 𝑀 = π‘Š)
98fveq2d 6888 . . . . . . 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 2784 . . . . . 6 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (Baseβ€˜π‘€) = 𝐡)
12 simpr 484 . . . . . 6 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ π‘Ž = 𝐴)
1311, 12oveq12d 7422 . . . . 5 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ ((Baseβ€˜π‘€) ↑m π‘Ž) = (𝐡 ↑m 𝐴))
14 simpll 764 . . . . . . . . . . 11 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ 𝑣 = 𝑉)
1514fveq2d 6888 . . . . . . . . . 10 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (leβ€˜π‘£) = (leβ€˜π‘‰))
16 mntoval.3 . . . . . . . . . 10 ≀ = (leβ€˜π‘‰)
1715, 16eqtr4di 2784 . . . . . . . . 9 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (leβ€˜π‘£) = ≀ )
1817breqd 5152 . . . . . . . 8 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (π‘₯(leβ€˜π‘£)𝑦 ↔ π‘₯ ≀ 𝑦))
198fveq2d 6888 . . . . . . . . . 10 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (leβ€˜π‘€) = (leβ€˜π‘Š))
20 mntoval.4 . . . . . . . . . 10 ≲ = (leβ€˜π‘Š)
2119, 20eqtr4di 2784 . . . . . . . . 9 (((𝑣 = 𝑉 ∧ 𝑀 = π‘Š) ∧ π‘Ž = 𝐴) β†’ (leβ€˜π‘€) = ≲ )
2221breqd 5152 . . . . . . . 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 344 . . . . . . 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 3336 . . . . . 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 3336 . . . . 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 3443 . . . 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 3928 . . 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 481 . 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 3487 . . 3 (𝑉 ∈ 𝑋 β†’ 𝑉 ∈ V)
3029adantr 480 . 2 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ 𝑉 ∈ V)
31 elex 3487 . . 3 (π‘Š ∈ π‘Œ β†’ π‘Š ∈ V)
3231adantl 481 . 2 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ π‘Š ∈ V)
33 eqid 2726 . . 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 7439 . . 3 ((𝑉 ∈ 𝑋 ∧ π‘Š ∈ π‘Œ) β†’ (𝐡 ↑m 𝐴) ∈ V)
3533, 34rabexd 5326 . 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 7555 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 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  {crab 3426  Vcvv 3468  β¦‹csb 3888   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406   ↑m cmap 8819  Basecbs 17150  lecple 17210  Monotcmnt 32650
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-pr 5420
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-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-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-mnt 32652
This theorem is referenced by:  ismnt  32655
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