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Theorem dssmapfv3d 38630
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 38629 . . 3 (𝜑𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵𝑠)))))
8 difeq2 3755 . . . . . 6 (𝑠 = 𝑆 → (𝐵𝑠) = (𝐵𝑆))
98fveq2d 6233 . . . . 5 (𝑠 = 𝑆 → (𝐹‘(𝐵𝑠)) = (𝐹‘(𝐵𝑆)))
109difeq2d 3761 . . . 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)
13 difexg 4841 . . . 4 (𝐵𝑉 → (𝐵 ∖ (𝐹‘(𝐵𝑆))) ∈ V)
144, 13syl 17 . . 3 (𝜑 → (𝐵 ∖ (𝐹‘(𝐵𝑆))) ∈ V)
157, 11, 12, 14fvmptd 6327 . 2 (𝜑 → (𝐺𝑆) = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
161, 15syl5eq 2697 1 (𝜑𝑇 = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
Colors of variables: wff setvar class
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-pow 4873  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:  ntrclselnel1  38672  ntrclsfv  38674  ntrclscls00  38681  ntrclsiso  38682  ntrclsk2  38683  ntrclskb  38684  ntrclsk3  38685  ntrclsk13  38686  dssmapntrcls  38743
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