Theorem dssmapfvd 43512

Description: Value of the duality operator for self-mappings of subsets of a base set, 𝐵. (Contributed by RP, 19-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
dssmapfvd	⊢ (𝜑 → 𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))))

Distinct variable groups:   𝐵,𝑏,𝑓   𝐵,𝑠,𝑏   𝜑,𝑏

Allowed substitution hints:   𝜑(𝑓,𝑠)   𝐷(𝑓,𝑠,𝑏)   𝑂(𝑓,𝑠,𝑏)   𝑉(𝑓,𝑠,𝑏)

Proof of Theorem dssmapfvd

Step	Hyp	Ref	Expression
1		dssmapfvd.d	. 2 ⊢ 𝐷 = (𝑂‘𝐵)
2		dssmapfvd.o	. . 3 ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))
3		pweq 4617	. . . . 5 ⊢ (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
4	3, 3	oveq12d 7435	. . . 4 ⊢ (𝑏 = 𝐵 → (𝒫 𝑏 ↑m 𝒫 𝑏) = (𝒫 𝐵 ↑m 𝒫 𝐵))
5		id 22	. . . . . 6 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵)
6		difeq1 4112	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 ∖ 𝑠) = (𝐵 ∖ 𝑠))
7	6	fveq2d 6898	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑓‘(𝑏 ∖ 𝑠)) = (𝑓‘(𝐵 ∖ 𝑠)))
8	5, 7	difeq12d 4120	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))) = (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))
9	3, 8	mpteq12dv 5239	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠)))) = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))))
10	4, 9	mpteq12dv 5239	. . 3 ⊢ (𝑏 = 𝐵 → (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))))))
11		dssmapfvd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
12	11	elexd 3485	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
13		ovex 7450	. . . 4 ⊢ (𝒫 𝐵 ↑m 𝒫 𝐵) ∈ V
14		mptexg 7231	. . . 4 ⊢ ((𝒫 𝐵 ↑m 𝒫 𝐵) ∈ V → (𝑓 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) ∈ V)
15	13, 14	mp1i 13	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) ∈ V)
16	2, 10, 12, 15	fvmptd3 7025	. 2 ⊢ (𝜑 → (𝑂‘𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))))
17	1, 16	eqtrid 2777	1 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3463   ∖ cdif 3942  𝒫 cpw 4603   ↦ cmpt 5231  ‘cfv 6547  (class class class)co 7417   ↑_m cmap 8843

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 5285  ax-sep 5299  ax-nul 5306  ax-pr 5428

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 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7420

This theorem is referenced by:  dssmapfv2d  43513  dssmapnvod  43515  dssmapf1od  43516