Theorem dssmapfv3d 44728

Description: Value of the duality operator for self-mappings of subsets of a base set, 𝐵 when applied to function 𝐹 and subset 𝑆. (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)
dssmapfv2d.f	⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
dssmapfv2d.g	⊢ 𝐺 = (𝐷‘𝐹)
dssmapfv3d.s	⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
dssmapfv3d.t	⊢ 𝑇 = (𝐺‘𝑆)

Assertion

Ref	Expression
dssmapfv3d	⊢ (𝜑 → 𝑇 = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))))

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

Allowed substitution hints:   𝐷(𝑓,𝑠,𝑏)   𝑆(𝑓,𝑏)   𝑇(𝑓,𝑠,𝑏)   𝐹(𝑏)   𝐺(𝑓,𝑠,𝑏)   𝑂(𝑓,𝑠,𝑏)   𝑉(𝑓,𝑠,𝑏)

Proof of Theorem dssmapfv3d

Step	Hyp	Ref	Expression
1		dssmapfv3d.t	. 2 ⊢ 𝑇 = (𝐺‘𝑆)
2		dssmapfvd.o	. . . 4 ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))
3		dssmapfvd.d	. . . 4 ⊢ 𝐷 = (𝑂‘𝐵)
4		dssmapfvd.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
5		dssmapfv2d.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
6		dssmapfv2d.g	. . . 4 ⊢ 𝐺 = (𝐷‘𝐹)
7	2, 3, 4, 5, 6	dssmapfv2d 44727	. . 3 ⊢ (𝜑 → 𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠)))))
8		difeq2 4076	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝐵 ∖ 𝑠) = (𝐵 ∖ 𝑆))
9	8	fveq2d 6887	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝐹‘(𝐵 ∖ 𝑠)) = (𝐹‘(𝐵 ∖ 𝑆)))
10	9	difeq2d 4082	. . . 4 ⊢ (𝑠 = 𝑆 → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))))
11	10	adantl 486	. . 3 ⊢ ((𝜑 ∧ 𝑠 = 𝑆) → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))))
12		dssmapfv3d.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
13	4	difexd 5303	. . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))) ∈ V)
14	7, 11, 12, 13	fvmptd 6999	. 2 ⊢ (𝜑 → (𝐺‘𝑆) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))))
15	1, 14	eqtrid 2810	1 ⊢ (𝜑 → 𝑇 = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3903  𝒫 cpw 4563   ↦ cmpt 5193  ‘cfv 6538  (class class class)co 7412   ↑_m cmap 8825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415

This theorem is referenced by:  ntrclselnel1  44766  ntrclsfv  44768  ntrclscls00  44775  ntrclsiso  44776  ntrclsk2  44777  ntrclskb  44778  ntrclsk3  44779  ntrclsk13  44780  dssmapntrcls  44837