Theorem dssmapfvd 38628

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	. . . 4 ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))
3	2	a1i 11	. . 3 ⊢ (𝜑 → 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠)))))))
4		pweq 4194	. . . . . 6 ⊢ (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
5	4, 4	oveq12d 6708	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝒫 𝑏 ↑𝑚 𝒫 𝑏) = (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
6		id 22	. . . . . . 7 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵)
7		difeq1 3754	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑏 ∖ 𝑠) = (𝐵 ∖ 𝑠))
8	7	fveq2d 6233	. . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑓‘(𝑏 ∖ 𝑠)) = (𝑓‘(𝐵 ∖ 𝑠)))
9	6, 8	difeq12d 3762	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))) = (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))
10	4, 9	mpteq12dv 4766	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠)))) = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))))
11	5, 10	mpteq12dv 4766	. . . 4 ⊢ (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))))))
12	11	adantl 481	. . 3 ⊢ ((𝜑 ∧ 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))))))
13		dssmapfvd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
14	13	elexd 3245	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
15		ovex 6718	. . . 4 ⊢ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∈ V
16		mptexg 6525	. . . 4 ⊢ ((𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∈ V → (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) ∈ V)
17	15, 16	mp1i 13	. . 3 ⊢ (𝜑 → (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) ∈ V)
18	3, 12, 14, 17	fvmptd 6327	. 2 ⊢ (𝜑 → (𝑂‘𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))))
19	1, 18	syl5eq 2697	1 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))))

Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∖ cdif 3604  𝒫 cpw 4191   ↦ cmpt 4762  ‘cfv 5926  (class class class)co 6690   ↑_𝑚 cmap 7899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693

This theorem is referenced by:  dssmapfv2d  38629  dssmapnvod  38631  dssmapf1od  38632