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Theorem mntoval 31842
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 31840 . . 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 6857 . . . 4 ((𝑣 = 𝑉𝑤 = 𝑊) → (Base‘𝑣) ∈ V)
4 fveq2 6842 . . . . . 6 (𝑣 = 𝑉 → (Base‘𝑣) = (Base‘𝑉))
5 mntoval.1 . . . . . 6 𝐴 = (Base‘𝑉)
64, 5eqtr4di 2794 . . . . 5 (𝑣 = 𝑉 → (Base‘𝑣) = 𝐴)
76adantr 481 . . . 4 ((𝑣 = 𝑉𝑤 = 𝑊) → (Base‘𝑣) = 𝐴)
8 simplr 767 . . . . . . . 8 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑤 = 𝑊)
98fveq2d 6846 . . . . . . 7 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = (Base‘𝑊))
10 mntoval.2 . . . . . . 7 𝐵 = (Base‘𝑊)
119, 10eqtr4di 2794 . . . . . 6 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (Base‘𝑤) = 𝐵)
12 simpr 485 . . . . . 6 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
1311, 12oveq12d 7375 . . . . 5 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → ((Base‘𝑤) ↑m 𝑎) = (𝐵m 𝐴))
14 simpll 765 . . . . . . . . . . 11 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → 𝑣 = 𝑉)
1514fveq2d 6846 . . . . . . . . . 10 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑣) = (le‘𝑉))
16 mntoval.3 . . . . . . . . . 10 = (le‘𝑉)
1715, 16eqtr4di 2794 . . . . . . . . 9 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑣) = )
1817breqd 5116 . . . . . . . 8 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (𝑥(le‘𝑣)𝑦𝑥 𝑦))
198fveq2d 6846 . . . . . . . . . 10 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑤) = (le‘𝑊))
20 mntoval.4 . . . . . . . . . 10 = (le‘𝑊)
2119, 20eqtr4di 2794 . . . . . . . . 9 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (le‘𝑤) = )
2221breqd 5116 . . . . . . . 8 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → ((𝑓𝑥)(le‘𝑤)(𝑓𝑦) ↔ (𝑓𝑥) (𝑓𝑦)))
2318, 22imbi12d 344 . . . . . . 7 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → ((𝑥(le‘𝑣)𝑦 → (𝑓𝑥)(le‘𝑤)(𝑓𝑦)) ↔ (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))))
2412, 23raleqbidv 3319 . . . . . 6 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (∀𝑦𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓𝑥)(le‘𝑤)(𝑓𝑦)) ↔ ∀𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))))
2512, 24raleqbidv 3319 . . . . 5 (((𝑣 = 𝑉𝑤 = 𝑊) ∧ 𝑎 = 𝐴) → (∀𝑥𝑎𝑦𝑎 (𝑥(le‘𝑣)𝑦 → (𝑓𝑥)(le‘𝑤)(𝑓𝑦)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))))
2613, 25rabeqbidv 3424 . . . 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 3895 . . 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 482 . 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 3463 . . 3 (𝑉𝑋𝑉 ∈ V)
3029adantr 481 . 2 ((𝑉𝑋𝑊𝑌) → 𝑉 ∈ V)
31 elex 3463 . . 3 (𝑊𝑌𝑊 ∈ V)
3231adantl 482 . 2 ((𝑉𝑋𝑊𝑌) → 𝑊 ∈ V)
33 eqid 2736 . . 3 {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))} = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))}
34 ovexd 7392 . . 3 ((𝑉𝑋𝑊𝑌) → (𝐵m 𝐴) ∈ V)
3533, 34rabexd 5290 . 2 ((𝑉𝑋𝑊𝑌) → {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))} ∈ V)
362, 28, 30, 32, 35ovmpod 7507 1 ((𝑉𝑋𝑊𝑌) → (𝑉Monot𝑊) = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦𝐴 (𝑥 𝑦 → (𝑓𝑥) (𝑓𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445  csb 3855   class class class wbr 5105  cfv 6496  (class class class)co 7357  cmpo 7359  m cmap 8765  Basecbs 17083  lecple 17140  Monotcmnt 31838
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-mnt 31840
This theorem is referenced by:  ismnt  31843
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