Theorem dssmapf1od 43348

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 43344	. . 3 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))))
5	3	pwexd 5370	. . . . . 6 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
6	5	mptexd 7221	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) ∈ V)
7	6	ralrimivw 3144	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)(𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) ∈ V)
8		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑓(𝒫 𝐵 ↑m 𝒫 𝐵)
9	8	fnmptf 6680	. . . 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))
10	7, 9	syl 17	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) Fn (𝒫 𝐵 ↑m 𝒫 𝐵))
11		fneq1 6634	. . . 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)))
12	11	biimprd 247	. . 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)))
13	4, 10, 12	sylc 65	. 2 ⊢ (𝜑 → 𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵))
14	1, 2, 3	dssmapnvod 43347	. 2 ⊢ (𝜑 → ◡𝐷 = 𝐷)
15		nvof1o 7274	. 2 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ ◡𝐷 = 𝐷) → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
16	13, 14, 15	syl2anc 583	1 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ∀wral 3055  Vcvv 3468   ∖ cdif 3940  𝒫 cpw 4597   ↦ cmpt 5224  ^◡ccnv 5668   Fn wfn 6532  –1-1-onto→wf1o 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:  dssmap2d  43349  ntrclsf1o  43378  clsneif1o  43431  clsneikex  43433  clsneinex  43434  clsneiel1  43435  neicvgf1o  43441  neicvgmex  43444  neicvgel1  43446  dssmapntrcls  43455  dssmapclsntr  43456