Theorem dssmap2d 43995

Description: For any base set 𝐵 the duality operator for self-mappings of subsets of that base set when composed with itself is the restricted identity operator. (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
dssmap2d	⊢ (𝜑 → (𝐷 ∘ 𝐷) = ( I ↾ (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 dssmap2d

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	dssmapnvod 43993	. . 3 ⊢ (𝜑 → ◡𝐷 = 𝐷)
5	4	coeq1d 5808	. 2 ⊢ (𝜑 → (◡𝐷 ∘ 𝐷) = (𝐷 ∘ 𝐷))
6	1, 2, 3	dssmapf1od 43994	. . 3 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
7		f1ococnv1 6797	. . 3 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) → (◡𝐷 ∘ 𝐷) = ( I ↾ (𝒫 𝐵 ↑m 𝒫 𝐵)))
8	6, 7	syl 17	. 2 ⊢ (𝜑 → (◡𝐷 ∘ 𝐷) = ( I ↾ (𝒫 𝐵 ↑m 𝒫 𝐵)))
9	5, 8	eqtr3d 2766	1 ⊢ (𝜑 → (𝐷 ∘ 𝐷) = ( I ↾ (𝒫 𝐵 ↑m 𝒫 𝐵)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3438   ∖ cdif 3902  𝒫 cpw 4553   ↦ cmpt 5176   I cid 5517  ^◡ccnv 5622   ↾ cres 5625   ∘ ccom 5627  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7353   ↑_m cmap 8760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-map 8762

This theorem is referenced by: (None)