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Theorem mntoval 30675
Description: Operation value of the monotone function. (Contributed by Thierry Arnoux, 23-Apr-2024.)
Hypotheses
Ref Expression
mntoval.1 𝐴 = (Base‘𝑉)
mntoval.2 𝐵 = (Base‘𝑊)
mntoval.3 = (le‘𝑉)
mntoval.4 = (le‘𝑊)
Assertion
Ref Expression
mntoval ((𝑉𝑋𝑊𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))})
Distinct variable groups:   𝑥,𝐴,𝑦,𝑓   𝐵,𝑓   𝑓,𝑉,𝑥,𝑦   𝑥,𝑊,𝑦,𝑓
Allowed substitution hints:   𝐵(𝑥,𝑦)   (𝑥,𝑦,𝑓)   𝑋(𝑥,𝑦,𝑓)   𝑌(𝑥,𝑦,𝑓)   (𝑥,𝑦,𝑓)

Proof of Theorem mntoval
Dummy variables 𝑎 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mnt 30673 . . 3 Monot = (𝑣 ∈ V, 𝑤 ∈ V ↦ (Base‘𝑣) / 𝑎{𝑓 ∈ ((Base‘𝑤) ↑m 𝑎) ∣ ∀𝑥𝑎𝑦𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓𝑥)(le‘𝑤)(𝑓𝑦))})
21a1i 11 . 2 ((𝑉𝑋𝑊𝑌) → Monot = (𝑣 ∈ V, 𝑤 ∈ V ↦ (Base‘𝑣) / 𝑎{𝑓 ∈ ((Base‘𝑤) ↑m 𝑎) ∣ ∀𝑥𝑎𝑦𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓𝑥)(le‘𝑤)(𝑓𝑦))}))
3 fvexd 6676 . . . 4 ((𝑣 = 𝑉𝑤 = 𝑊) → (Base‘𝑣) ∈ V)
4 fveq2 6661 . . . . . 6 (𝑣 = 𝑉 → (Base‘𝑣) = (Base‘𝑉))
5 mntoval.1 . . . . . 6 𝐴 = (Base‘𝑉)
64, 5syl6eqr 2877 . . . . 5 (𝑣 = 𝑉 → (Base‘𝑣) = 𝐴)
76adantr 484 . . . 4 ((𝑣 = 𝑉𝑤 = 𝑊) → (Base‘𝑣) = 𝐴)
8 simplr 768 . . . . . . . 8 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑤 = 𝑊)
98fveq2d 6665 . . . . . . 7 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
10 mntoval.2 . . . . . . 7 𝐵 = (Base‘𝑊)
119, 10syl6eqr 2877 . . . . . 6 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
12 simpr 488 . . . . . 6 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
1311, 12oveq12d 7167 . . . . 5 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → ((Base‘𝑤) ↑m 𝑎) = (𝐵m 𝐴))
14 simpll 766 . . . . . . . . . . 11 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑣 = 𝑉)
1514fveq2d 6665 . . . . . . . . . 10 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑣) = (le‘𝑉))
16 mntoval.3 . . . . . . . . . 10 = (le‘𝑉)
1715, 16syl6eqr 2877 . . . . . . . . 9 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑣) = )
1817breqd 5063 . . . . . . . 8 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (𝑥(le‘𝑣)𝑦𝑥 𝑦))
198fveq2d 6665 . . . . . . . . . 10 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑤) = (le‘𝑊))
20 mntoval.4 . . . . . . . . . 10 = (le‘𝑊)
2119, 20syl6eqr 2877 . . . . . . . . 9 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑤) = )
2221breqd 5063 . . . . . . . 8 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → ((𝑓𝑥)(le‘𝑤)(𝑓𝑦) ↔ (𝑓𝑥) (𝑓𝑦)))
2318, 22imbi12d 348 . . . . . . 7 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → ((𝑥(le‘𝑣)𝑦 → (𝑓𝑥)(le‘𝑤)(𝑓𝑦)) ↔ (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))))
2412, 23raleqbidv 3392 . . . . . 6 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (∀𝑦𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓𝑥)(le‘𝑤)(𝑓𝑦)) ↔ ∀𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))))
2512, 24raleqbidv 3392 . . . . 5 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (∀𝑥𝑎𝑦𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓𝑥)(le‘𝑤)(𝑓𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))))
2613, 25rabeqbidv 3471 . . . 4 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → {𝑓 ∈ ((Base‘𝑤) ↑m 𝑎) ∣ ∀𝑥𝑎𝑦𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓𝑥)(le‘𝑤)(𝑓𝑦))} = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))})
273, 7, 26csbied2 3903 . . 3 ((𝑣 = 𝑉𝑤 = 𝑊) → (Base‘𝑣) / 𝑎{𝑓 ∈ ((Base‘𝑤) ↑m 𝑎) ∣ ∀𝑥𝑎𝑦𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓𝑥)(le‘𝑤)(𝑓𝑦))} = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))})
2827adantl 485 . 2 (((𝑉𝑋𝑊𝑌) ∧ (𝑣 = 𝑉𝑤 = 𝑊)) → (Base‘𝑣) / 𝑎{𝑓 ∈ ((Base‘𝑤) ↑m 𝑎) ∣ ∀𝑥𝑎𝑦𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓𝑥)(le‘𝑤)(𝑓𝑦))} = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))})
29 elex 3498 . . 3 (𝑉𝑋𝑉 ∈ V)
3029adantr 484 . 2 ((𝑉𝑋𝑊𝑌) → 𝑉 ∈ V)
31 elex 3498 . . 3 (𝑊𝑌𝑊 ∈ V)
3231adantl 485 . 2 ((𝑉𝑋𝑊𝑌) → 𝑊 ∈ V)
33 eqid 2824 . . 3 {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))} = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))}
34 ovexd 7184 . . 3 ((𝑉𝑋𝑊𝑌) → (𝐵m 𝐴) ∈ V)
3533, 34rabexd 5222 . 2 ((𝑉𝑋𝑊𝑌) → {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))} ∈ V)
362, 28, 30, 32, 35ovmpod 7295 1 ((𝑉𝑋𝑊𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  {crab 3137  Vcvv 3480  csb 3866   class class class wbr 5052  cfv 6343  (class class class)co 7149  cmpo 7151  m cmap 8402  Basecbs 16483  lecple 16572  Monotcmnt 30671
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-mnt 30673
This theorem is referenced by:  ismnt  30676
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