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Theorem dssmapfv3d 43591
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 43590 . . 3 (𝜑𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵𝑠)))))
8 difeq2 4112 . . . . . 6 (𝑠 = 𝑆 → (𝐵𝑠) = (𝐵𝑆))
98fveq2d 6900 . . . . 5 (𝑠 = 𝑆 → (𝐹‘(𝐵𝑠)) = (𝐹‘(𝐵𝑆)))
109difeq2d 4118 . . . 4 (𝑠 = 𝑆 → (𝐵 ∖ (𝐹‘(𝐵𝑠))) = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
1110adantl 480 . . 3 ((𝜑𝑠 = 𝑆) → (𝐵 ∖ (𝐹‘(𝐵𝑠))) = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
12 dssmapfv3d.s . . 3 (𝜑𝑆 ∈ 𝒫 𝐵)
134difexd 5332 . . 3 (𝜑 → (𝐵 ∖ (𝐹‘(𝐵𝑆))) ∈ V)
147, 11, 12, 13fvmptd 7011 . 2 (𝜑 → (𝐺𝑆) = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
151, 14eqtrid 2777 1 (𝜑𝑇 = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3461  cdif 3941  𝒫 cpw 4604  cmpt 5232  cfv 6549  (class class class)co 7419  m cmap 8845
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429
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 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422
This theorem is referenced by:  ntrclselnel1  43629  ntrclsfv  43631  ntrclscls00  43638  ntrclsiso  43639  ntrclsk2  43640  ntrclskb  43641  ntrclsk3  43642  ntrclsk13  43643  dssmapntrcls  43700
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