Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dssmapfvd Structured version   Visualization version   GIF version

Theorem dssmapfvd 41625
Description: Value of the duality operator for self-mappings of subsets of a base set, 𝐵. (Contributed by RP, 19-Apr-2021.)
Hypotheses
Ref Expression
dssmapfvd.o 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))
dssmapfvd.d 𝐷 = (𝑂𝐵)
dssmapfvd.b (𝜑𝐵𝑉)
Assertion
Ref Expression
dssmapfvd (𝜑𝐷 = (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
Distinct variable groups:   𝐵,𝑏,𝑓   𝐵,𝑠,𝑏   𝜑,𝑏
Allowed substitution hints:   𝜑(𝑓,𝑠)   𝐷(𝑓,𝑠,𝑏)   𝑂(𝑓,𝑠,𝑏)   𝑉(𝑓,𝑠,𝑏)

Proof of Theorem dssmapfvd
StepHypRef Expression
1 dssmapfvd.d . 2 𝐷 = (𝑂𝐵)
2 dssmapfvd.o . . 3 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))
3 pweq 4549 . . . . 5 (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
43, 3oveq12d 7293 . . . 4 (𝑏 = 𝐵 → (𝒫 𝑏m 𝒫 𝑏) = (𝒫 𝐵m 𝒫 𝐵))
5 id 22 . . . . . 6 (𝑏 = 𝐵𝑏 = 𝐵)
6 difeq1 4050 . . . . . . 7 (𝑏 = 𝐵 → (𝑏𝑠) = (𝐵𝑠))
76fveq2d 6778 . . . . . 6 (𝑏 = 𝐵 → (𝑓‘(𝑏𝑠)) = (𝑓‘(𝐵𝑠)))
85, 7difeq12d 4058 . . . . 5 (𝑏 = 𝐵 → (𝑏 ∖ (𝑓‘(𝑏𝑠))) = (𝐵 ∖ (𝑓‘(𝐵𝑠))))
93, 8mpteq12dv 5165 . . . 4 (𝑏 = 𝐵 → (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠)))) = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))
104, 9mpteq12dv 5165 . . 3 (𝑏 = 𝐵 → (𝑓 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))) = (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
11 dssmapfvd.b . . . 4 (𝜑𝐵𝑉)
1211elexd 3452 . . 3 (𝜑𝐵 ∈ V)
13 ovex 7308 . . . 4 (𝒫 𝐵m 𝒫 𝐵) ∈ V
14 mptexg 7097 . . . 4 ((𝒫 𝐵m 𝒫 𝐵) ∈ V → (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ∈ V)
1513, 14mp1i 13 . . 3 (𝜑 → (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ∈ V)
162, 10, 12, 15fvmptd3 6898 . 2 (𝜑 → (𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
171, 16eqtrid 2790 1 (𝜑𝐷 = (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  Vcvv 3432  cdif 3884  𝒫 cpw 4533  cmpt 5157  cfv 6433  (class class class)co 7275  m cmap 8615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278
This theorem is referenced by:  dssmapfv2d  41626  dssmapnvod  41628  dssmapf1od  41629
  Copyright terms: Public domain W3C validator