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Theorem dssmapf1od 43593
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 43589 . . 3 (𝜑𝐷 = (𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
53pwexd 5379 . . . . . 6 (𝜑 → 𝒫 𝐵 ∈ V)
65mptexd 7236 . . . . 5 (𝜑 → (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))) ∈ V)
76ralrimivw 3139 . . . 4 (𝜑 → ∀𝑓 ∈ (𝒫 𝐵m 𝒫 𝐵)(𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))) ∈ V)
8 nfcv 2891 . . . . 5 𝑓(𝒫 𝐵m 𝒫 𝐵)
98fnmptf 6692 . . . 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 6646 . . . 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 247 . . 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 43592 . 2 (𝜑𝐷 = 𝐷)
15 nvof1o 7289 . 2 ((𝐷 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ 𝐷 = 𝐷) → 𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
1613, 14, 15syl2anc 582 1 (𝜑𝐷:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵m 𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wral 3050  Vcvv 3461  cdif 3941  𝒫 cpw 4604  cmpt 5232  ccnv 5677   Fn wfn 6544  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  m cmap 8845
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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847
This theorem is referenced by:  dssmap2d  43594  ntrclsf1o  43623  clsneif1o  43676  clsneikex  43678  clsneinex  43679  clsneiel1  43680  neicvgf1o  43686  neicvgmex  43689  neicvgel1  43691  dssmapntrcls  43700  dssmapclsntr  43701
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