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Theorem dssmap2d 43346
Description: For any base set 𝐵 the duality operator for self-mappings of subsets of that base set when composed with itself is the restricted identity operator. (Contributed by RP, 21-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
dssmap2d (𝜑 → (𝐷𝐷) = ( I ↾ (𝒫 𝐵m 𝒫 𝐵)))
Distinct variable groups:   𝐵,𝑏,𝑓,𝑠   𝜑,𝑏,𝑓,𝑠
Allowed substitution hints:   𝐷(𝑓,𝑠,𝑏)   𝑂(𝑓,𝑠,𝑏)   𝑉(𝑓,𝑠,𝑏)

Proof of Theorem dssmap2d
StepHypRef Expression
1 dssmapfvd.o . . . 4 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))
2 dssmapfvd.d . . . 4 𝐷 = (𝑂𝐵)
3 dssmapfvd.b . . . 4 (𝜑𝐵𝑉)
41, 2, 3dssmapnvod 43344 . . 3 (𝜑𝐷 = 𝐷)
54coeq1d 5855 . 2 (𝜑 → (𝐷𝐷) = (𝐷𝐷))
61, 2, 3dssmapf1od 43345 . . 3 (𝜑𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
7 f1ococnv1 6856 . . 3 (𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵) → (𝐷𝐷) = ( I ↾ (𝒫 𝐵m 𝒫 𝐵)))
86, 7syl 17 . 2 (𝜑 → (𝐷𝐷) = ( I ↾ (𝒫 𝐵m 𝒫 𝐵)))
95, 8eqtr3d 2768 1 (𝜑 → (𝐷𝐷) = ( I ↾ (𝒫 𝐵m 𝒫 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3468  cdif 3940  𝒫 cpw 4597  cmpt 5224   I cid 5566  ccnv 5668  cres 5671  ccom 5673  1-1-ontowf1o 6536  cfv 6537  (class class class)co 7405  m cmap 8822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-map 8824
This theorem is referenced by: (None)
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