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Theorem dssmapfvd 40241
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 4538 . . . . 5 (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
43, 3oveq12d 7163 . . . 4 (𝑏 = 𝐵 → (𝒫 𝑏m 𝒫 𝑏) = (𝒫 𝐵m 𝒫 𝐵))
5 id 22 . . . . . 6 (𝑏 = 𝐵𝑏 = 𝐵)
6 difeq1 4089 . . . . . . 7 (𝑏 = 𝐵 → (𝑏𝑠) = (𝐵𝑠))
76fveq2d 6667 . . . . . 6 (𝑏 = 𝐵 → (𝑓‘(𝑏𝑠)) = (𝑓‘(𝐵𝑠)))
85, 7difeq12d 4097 . . . . 5 (𝑏 = 𝐵 → (𝑏 ∖ (𝑓‘(𝑏𝑠))) = (𝐵 ∖ (𝑓‘(𝐵𝑠))))
93, 8mpteq12dv 5142 . . . 4 (𝑏 = 𝐵 → (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠)))) = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))
104, 9mpteq12dv 5142 . . 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 3512 . . 3 (𝜑𝐵 ∈ V)
13 ovex 7178 . . . 4 (𝒫 𝐵m 𝒫 𝐵) ∈ V
14 mptexg 6975 . . . 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 6783 . 2 (𝜑 → (𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
171, 16syl5eq 2865 1 (𝜑𝐷 = (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  cdif 3930  𝒫 cpw 4535  cmpt 5137  cfv 6348  (class class class)co 7145  m cmap 8395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148
This theorem is referenced by:  dssmapfv2d  40242  dssmapnvod  40244  dssmapf1od  40245
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