Theorem dssmapfv3d 43992

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 43991	. . 3 ⊢ (𝜑 → 𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠)))))
8		difeq2 4073	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝐵 ∖ 𝑠) = (𝐵 ∖ 𝑆))
9	8	fveq2d 6830	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝐹‘(𝐵 ∖ 𝑠)) = (𝐹‘(𝐵 ∖ 𝑆)))
10	9	difeq2d 4079	. . . 4 ⊢ (𝑠 = 𝑆 → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))))
11	10	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑠 = 𝑆) → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))))
12		dssmapfv3d.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
13	4	difexd 5273	. . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))) ∈ V)
14	7, 11, 12, 13	fvmptd 6941	. 2 ⊢ (𝜑 → (𝐺‘𝑆) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))))
15	1, 14	eqtrid 2776	1 ⊢ (𝜑 → 𝑇 = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3438   ∖ cdif 3902  𝒫 cpw 4553   ↦ cmpt 5176  ‘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

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

This theorem is referenced by:  ntrclselnel1  44030  ntrclsfv  44032  ntrclscls00  44039  ntrclsiso  44040  ntrclsk2  44041  ntrclskb  44042  ntrclsk3  44043  ntrclsk13  44044  dssmapntrcls  44101