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Theorem dssmapf1od 43923
Description: For any base set 𝐵 the duality operator for self-mappings of subsets of that base set is one-to-one and onto. (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
dssmapf1od (𝜑𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(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 dssmapf1od
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, 3dssmapfvd 43919 . . 3 (𝜑𝐷 = (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
53pwexd 5400 . . . . . 6 (𝜑 → 𝒫 𝐵 ∈ V)
65mptexd 7259 . . . . 5 (𝜑 → (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))) ∈ V)
76ralrimivw 3152 . . . 4 (𝜑 → ∀𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵)(𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))) ∈ V)
8 nfcv 2904 . . . . 5 𝑓(𝒫 𝐵m 𝒫 𝐵)
98fnmptf 6715 . . . 4 (∀𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵)(𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))) ∈ V → (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) Fn (𝒫 𝐵m 𝒫 𝐵))
107, 9syl 17 . . 3 (𝜑 → (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) Fn (𝒫 𝐵m 𝒫 𝐵))
11 fneq1 6669 . . . 4 (𝐷 = (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) → (𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) ↔ (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) Fn (𝒫 𝐵m 𝒫 𝐵)))
1211biimprd 248 . . 3 (𝐷 = (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) → ((𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) Fn (𝒫 𝐵m 𝒫 𝐵) → 𝐷 Fn (𝒫 𝐵m 𝒫 𝐵)))
134, 10, 12sylc 65 . 2 (𝜑𝐷 Fn (𝒫 𝐵m 𝒫 𝐵))
141, 2, 3dssmapnvod 43922 . 2 (𝜑𝐷 = 𝐷)
15 nvof1o 7314 . 2 ((𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐷 = 𝐷) → 𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
1613, 14, 15syl2anc 583 1 (𝜑𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2103  wral 3063  Vcvv 3482  cdif 3967  𝒫 cpw 4622  cmpt 5252  ccnv 5698   Fn wfn 6567  1-1-ontowf1o 6571  cfv 6572  (class class class)co 7445  m cmap 8880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-1st 8026  df-2nd 8027  df-map 8882
This theorem is referenced by:  dssmap2d  43924  ntrclsf1o  43953  clsneif1o  44006  clsneikex  44008  clsneinex  44009  clsneiel1  44010  neicvgf1o  44016  neicvgmex  44019  neicvgel1  44021  dssmapntrcls  44030  dssmapclsntr  44031
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