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Theorem dssmapfv3d 40358
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 40357 . . 3 (𝜑𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵𝑠)))))
8 difeq2 4092 . . . . . 6 (𝑠 = 𝑆 → (𝐵𝑠) = (𝐵𝑆))
98fveq2d 6668 . . . . 5 (𝑠 = 𝑆 → (𝐹‘(𝐵𝑠)) = (𝐹‘(𝐵𝑆)))
109difeq2d 4098 . . . 4 (𝑠 = 𝑆 → (𝐵 ∖ (𝐹‘(𝐵𝑠))) = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
1110adantl 484 . . 3 ((𝜑𝑠 = 𝑆) → (𝐵 ∖ (𝐹‘(𝐵𝑠))) = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
12 dssmapfv3d.s . . 3 (𝜑𝑆 ∈ 𝒫 𝐵)
13 difexg 5223 . . . 4 (𝐵𝑉 → (𝐵 ∖ (𝐹‘(𝐵𝑆))) ∈ V)
144, 13syl 17 . . 3 (𝜑 → (𝐵 ∖ (𝐹‘(𝐵𝑆))) ∈ V)
157, 11, 12, 14fvmptd 6769 . 2 (𝜑 → (𝐺𝑆) = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
161, 15syl5eq 2868 1 (𝜑𝑇 = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2110  Vcvv 3494  cdif 3932  𝒫 cpw 4538  cmpt 5138  cfv 6349  (class class class)co 7150  m cmap 8400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153
This theorem is referenced by:  ntrclselnel1  40400  ntrclsfv  40402  ntrclscls00  40409  ntrclsiso  40410  ntrclsk2  40411  ntrclskb  40412  ntrclsk3  40413  ntrclsk13  40414  dssmapntrcls  40471
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