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Theorem dssmapfv3d 44032
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 𝑏m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))
dssmapfvd.d 𝐷 = (𝑂𝐵)
dssmapfvd.b (𝜑𝐵𝑉)
dssmapfv2d.f (𝜑𝐹 ∈ (𝒫 𝐵m 𝒫 𝐵))
dssmapfv2d.g 𝐺 = (𝐷𝐹)
dssmapfv3d.s (𝜑𝑆 ∈ 𝒫 𝐵)
dssmapfv3d.t 𝑇 = (𝐺𝑆)
Assertion
Ref Expression
dssmapfv3d (𝜑𝑇 = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
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
StepHypRef Expression
1 dssmapfv3d.t . 2 𝑇 = (𝐺𝑆)
2 dssmapfvd.o . . . 4 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))
3 dssmapfvd.d . . . 4 𝐷 = (𝑂𝐵)
4 dssmapfvd.b . . . 4 (𝜑𝐵𝑉)
5 dssmapfv2d.f . . . 4 (𝜑𝐹 ∈ (𝒫 𝐵m 𝒫 𝐵))
6 dssmapfv2d.g . . . 4 𝐺 = (𝐷𝐹)
72, 3, 4, 5, 6dssmapfv2d 44031 . . 3 (𝜑𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵𝑠)))))
8 difeq2 4120 . . . . . 6 (𝑠 = 𝑆 → (𝐵𝑠) = (𝐵𝑆))
98fveq2d 6910 . . . . 5 (𝑠 = 𝑆 → (𝐹‘(𝐵𝑠)) = (𝐹‘(𝐵𝑆)))
109difeq2d 4126 . . . 4 (𝑠 = 𝑆 → (𝐵 ∖ (𝐹‘(𝐵𝑠))) = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
1110adantl 481 . . 3 ((𝜑𝑠 = 𝑆) → (𝐵 ∖ (𝐹‘(𝐵𝑠))) = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
12 dssmapfv3d.s . . 3 (𝜑𝑆 ∈ 𝒫 𝐵)
134difexd 5331 . . 3 (𝜑 → (𝐵 ∖ (𝐹‘(𝐵𝑆))) ∈ V)
147, 11, 12, 13fvmptd 7023 . 2 (𝜑 → (𝐺𝑆) = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
151, 14eqtrid 2789 1 (𝜑𝑇 = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  𝒫 cpw 4600  cmpt 5225  cfv 6561  (class class class)co 7431  m cmap 8866
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434
This theorem is referenced by:  ntrclselnel1  44070  ntrclsfv  44072  ntrclscls00  44079  ntrclsiso  44080  ntrclsk2  44081  ntrclskb  44082  ntrclsk3  44083  ntrclsk13  44084  dssmapntrcls  44141
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