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Theorem dssmapfv3d 38313
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 38312 . . 3 (𝜑𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵𝑠)))))
8 difeq2 3722 . . . . . 6 (𝑠 = 𝑆 → (𝐵𝑠) = (𝐵𝑆))
98fveq2d 6195 . . . . 5 (𝑠 = 𝑆 → (𝐹‘(𝐵𝑠)) = (𝐹‘(𝐵𝑆)))
109difeq2d 3728 . . . 4 (𝑠 = 𝑆 → (𝐵 ∖ (𝐹‘(𝐵𝑠))) = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
1110adantl 482 . . 3 ((𝜑𝑠 = 𝑆) → (𝐵 ∖ (𝐹‘(𝐵𝑠))) = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
12 dssmapfv3d.s . . 3 (𝜑𝑆 ∈ 𝒫 𝐵)
13 difexg 4808 . . . 4 (𝐵𝑉 → (𝐵 ∖ (𝐹‘(𝐵𝑆))) ∈ V)
144, 13syl 17 . . 3 (𝜑 → (𝐵 ∖ (𝐹‘(𝐵𝑆))) ∈ V)
157, 11, 12, 14fvmptd 6288 . 2 (𝜑 → (𝐺𝑆) = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
161, 15syl5eq 2668 1 (𝜑𝑇 = (𝐵 ∖ (𝐹‘(𝐵𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1483  wcel 1990  Vcvv 3200  cdif 3571  𝒫 cpw 4158  cmpt 4729  cfv 5888  (class class class)co 6650  𝑚 cmap 7857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653
This theorem is referenced by:  ntrclselnel1  38355  ntrclsfv  38357  ntrclscls00  38364  ntrclsiso  38365  ntrclsk2  38366  ntrclskb  38367  ntrclsk3  38368  ntrclsk13  38369  dssmapntrcls  38426
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