Theorem dssmapf1od 38315

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 𝑏 ↑𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))
dssmapfvd.d	⊢ 𝐷 = (𝑂‘𝐵)
dssmapfvd.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
dssmapf1od	⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 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

Step	Hyp	Ref	Expression
1		dssmapfvd.o	. . . 4 ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))
2		dssmapfvd.d	. . . 4 ⊢ 𝐷 = (𝑂‘𝐵)
3		dssmapfvd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
4	1, 2, 3	dssmapfvd 38311	. . 3 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))))
5		pwexg 4850	. . . . . . 7 ⊢ (𝐵 ∈ 𝑉 → 𝒫 𝐵 ∈ V)
6	3, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
7		mptexg 6484	. . . . . 6 ⊢ (𝒫 𝐵 ∈ V → (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) ∈ V)
8	6, 7	syl 17	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) ∈ V)
9	8	ralrimivw 2967	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)(𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) ∈ V)
10		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑓(𝒫 𝐵 ↑𝑚 𝒫 𝐵)
11	10	fnmptf 6016	. . . 4 ⊢ (∀𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)(𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) ∈ V → (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
12	9, 11	syl 17	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
13		fneq1 5979	. . . 4 ⊢ (𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) → (𝐷 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↔ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵)))
14	13	biimprd 238	. . 3 ⊢ (𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) → ((𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → 𝐷 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵)))
15	4, 12, 14	sylc 65	. 2 ⊢ (𝜑 → 𝐷 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
16	1, 2, 3	dssmapnvod 38314	. 2 ⊢ (𝜑 → ◡𝐷 = 𝐷)
17		nvof1o 6536	. 2 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ ◡𝐷 = 𝐷) → 𝐷:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵))
18	15, 16, 17	syl2anc 693	1 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∖ cdif 3571  𝒫 cpw 4158   ↦ cmpt 4729  ^◡ccnv 5113   Fn wfn 5883  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  dssmap2d  38316  ntrclsf1o  38349  clsneif1o  38402  clsneikex  38404  clsneinex  38405  clsneiel1  38406  neicvgf1o  38412  neicvgmex  38415  neicvgel1  38417  dssmapntrcls  38426  dssmapclsntr  38427