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Mirrors > Home > MPE Home > Th. List > Mathboxes > dssmapfvd | Structured version Visualization version GIF version |
Description: Value of the duality operator for self-mappings of subsets of a base set, 𝐵. (Contributed by RP, 19-Apr-2021.) |
Ref | Expression |
---|---|
dssmapfvd.o | ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠)))))) |
dssmapfvd.d | ⊢ 𝐷 = (𝑂‘𝐵) |
dssmapfvd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
Ref | Expression |
---|---|
dssmapfvd | ⊢ (𝜑 → 𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dssmapfvd.d | . 2 ⊢ 𝐷 = (𝑂‘𝐵) | |
2 | dssmapfvd.o | . . . 4 ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠)))))) | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝜑 → 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))))) |
4 | pweq 4161 | . . . . . 6 ⊢ (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵) | |
5 | 4, 4 | oveq12d 6668 | . . . . 5 ⊢ (𝑏 = 𝐵 → (𝒫 𝑏 ↑𝑚 𝒫 𝑏) = (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
6 | id 22 | . . . . . . 7 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵) | |
7 | difeq1 3721 | . . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑏 ∖ 𝑠) = (𝐵 ∖ 𝑠)) | |
8 | 7 | fveq2d 6195 | . . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑓‘(𝑏 ∖ 𝑠)) = (𝑓‘(𝐵 ∖ 𝑠))) |
9 | 6, 8 | difeq12d 3729 | . . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))) = (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) |
10 | 4, 9 | mpteq12dv 4733 | . . . . 5 ⊢ (𝑏 = 𝐵 → (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠)))) = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) |
11 | 5, 10 | mpteq12dv 4733 | . . . 4 ⊢ (𝑏 = 𝐵 → (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))))) |
12 | 11 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑏 = 𝐵) → (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠))))) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))))) |
13 | dssmapfvd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
14 | 13 | elexd 3214 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
15 | ovex 6678 | . . . 4 ⊢ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∈ V | |
16 | mptexg 6484 | . . . 4 ⊢ ((𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∈ V → (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) ∈ V) | |
17 | 15, 16 | mp1i 13 | . . 3 ⊢ (𝜑 → (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) ∈ V) |
18 | 3, 12, 14, 17 | fvmptd 6288 | . 2 ⊢ (𝜑 → (𝑂‘𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))))) |
19 | 1, 18 | syl5eq 2668 | 1 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))))) |
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-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: dssmapfv2d 38312 dssmapnvod 38314 dssmapf1od 38315 |
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