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Theorem dssmapfvd 38831
Description: Value of the duality operator for self-mappings of subsets of a base set, 𝐵. (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)
Assertion
Ref Expression
dssmapfvd (𝜑𝐷 = (𝑓 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
Distinct variable groups:   𝐵,𝑏,𝑓   𝐵,𝑠,𝑏   𝜑,𝑏
Allowed substitution hints:   𝜑(𝑓,𝑠)   𝐷(𝑓,𝑠,𝑏)   𝑂(𝑓,𝑠,𝑏)   𝑉(𝑓,𝑠,𝑏)

Proof of Theorem dssmapfvd
StepHypRef Expression
1 dssmapfvd.d . 2 𝐷 = (𝑂𝐵)
2 dssmapfvd.o . . . 4 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))))
32a1i 11 . . 3 (𝜑𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠)))))))
4 pweq 4305 . . . . . 6 (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵)
54, 4oveq12d 6832 . . . . 5 (𝑏 = 𝐵 → (𝒫 𝑏𝑚 𝒫 𝑏) = (𝒫 𝐵𝑚 𝒫 𝐵))
6 id 22 . . . . . . 7 (𝑏 = 𝐵𝑏 = 𝐵)
7 difeq1 3864 . . . . . . . 8 (𝑏 = 𝐵 → (𝑏𝑠) = (𝐵𝑠))
87fveq2d 6357 . . . . . . 7 (𝑏 = 𝐵 → (𝑓‘(𝑏𝑠)) = (𝑓‘(𝐵𝑠)))
96, 8difeq12d 3872 . . . . . 6 (𝑏 = 𝐵 → (𝑏 ∖ (𝑓‘(𝑏𝑠))) = (𝐵 ∖ (𝑓‘(𝐵𝑠))))
104, 9mpteq12dv 4885 . . . . 5 (𝑏 = 𝐵 → (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠)))) = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠)))))
115, 10mpteq12dv 4885 . . . 4 (𝑏 = 𝐵 → (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))) = (𝑓 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
1211adantl 473 . . 3 ((𝜑𝑏 = 𝐵) → (𝑓 ∈ (𝒫 𝑏𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏𝑠))))) = (𝑓 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
13 dssmapfvd.b . . . 4 (𝜑𝐵𝑉)
1413elexd 3354 . . 3 (𝜑𝐵 ∈ V)
15 ovex 6842 . . . 4 (𝒫 𝐵𝑚 𝒫 𝐵) ∈ V
16 mptexg 6649 . . . 4 ((𝒫 𝐵𝑚 𝒫 𝐵) ∈ V → (𝑓 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ∈ V)
1715, 16mp1i 13 . . 3 (𝜑 → (𝑓 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))) ∈ V)
183, 12, 14, 17fvmptd 6451 . 2 (𝜑 → (𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
191, 18syl5eq 2806 1 (𝜑𝐷 = (𝑓 ∈ (𝒫 𝐵𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵𝑠))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cdif 3712  𝒫 cpw 4302  cmpt 4881  cfv 6049  (class class class)co 6814  𝑚 cmap 8025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817
This theorem is referenced by:  dssmapfv2d  38832  dssmapnvod  38834  dssmapf1od  38835
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