Theorem dssmapf1od 44448

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 44444	. . 3 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))))
5	3	pwexd 5321	. . . . . 6 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
6	5	mptexd 7179	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) ∈ V)
7	6	ralrimivw 3133	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)(𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) ∈ V)
8		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑓(𝒫 𝐵 ↑m 𝒫 𝐵)
9	8	fnmptf 6634	. . . 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 6589	. . . 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 248	. . 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 44447	. 2 ⊢ (𝜑 → ◡𝐷 = 𝐷)
15		nvof1o 7235	. 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 585	1 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ∖ cdif 3886  𝒫 cpw 4541   ↦ cmpt 5166  ^◡ccnv 5630   Fn wfn 6493  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775

This theorem is referenced by:  dssmap2d  44449  ntrclsf1o  44478  clsneif1o  44531  clsneikex  44533  clsneinex  44534  clsneiel1  44535  neicvgf1o  44541  neicvgmex  44544  neicvgel1  44546  dssmapntrcls  44555  dssmapclsntr  44556