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Theorem dssmapfvd 38628
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 . . . 4 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))
32a1i 11 . . 3 (𝜑𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠)))))))
4 pweq 4194 . . . . . 6 (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
54, 4oveq12d 6708 . . . . 5 (𝑏 = 𝐵 → (𝒫 𝑏𝑚 𝒫 𝑏) = (𝒫 𝐵𝑚 𝒫 𝐵))
6 id 22 . . . . . . 7 (𝑏 = 𝐵𝑏 = 𝐵)
7 difeq1 3754 . . . . . . . 8 (𝑏 = 𝐵 → (𝑏𝑠) = (𝐵𝑠))
87fveq2d 6233 . . . . . . 7 (𝑏 = 𝐵 → (𝑓‘(𝑏𝑠)) = (𝑓‘(𝐵𝑠)))
96, 8difeq12d 3762 . . . . . 6 (𝑏 = 𝐵 → (𝑏 ∖ (𝑓‘(𝑏𝑠))) = (𝐵 ∖ (𝑓‘(𝐵𝑠))))
104, 9mpteq12dv 4766 . . . . 5 (𝑏 = 𝐵 → (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠)))) = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))
115, 10mpteq12dv 4766 . . . 4 (𝑏 = 𝐵 → (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))) = (𝑓 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
1211adantl 481 . . 3 ((𝜑𝑏 = 𝐵) → (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))) = (𝑓 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
13 dssmapfvd.b . . . 4 (𝜑𝐵𝑉)
1413elexd 3245 . . 3 (𝜑𝐵 ∈ V)
15 ovex 6718 . . . 4 (𝒫 𝐵𝑚 𝒫 𝐵) ∈ V
16 mptexg 6525 . . . 4 ((𝒫 𝐵𝑚 𝒫 𝐵) ∈ V → (𝑓 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ∈ V)
1715, 16mp1i 13 . . 3 (𝜑 → (𝑓 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ∈ V)
183, 12, 14, 17fvmptd 6327 . 2 (𝜑 → (𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
191, 18syl5eq 2697 1 (𝜑𝐷 = (𝑓 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  𝒫 cpw 4191  cmpt 4762  cfv 5926  (class class class)co 6690  𝑚 cmap 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693
This theorem is referenced by:  dssmapfv2d  38629  dssmapnvod  38631  dssmapf1od  38632
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