|   | Mathbox for Richard Penner | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > dssmap2d | Structured version Visualization version GIF version | ||
| Description: For any base set 𝐵 the duality operator for self-mappings of subsets of that base set when composed with itself is the restricted identity operator. (Contributed by RP, 21-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 | 
|---|---|
| dssmap2d | ⊢ (𝜑 → (𝐷 ∘ 𝐷) = ( I ↾ (𝒫 𝐵 ↑m 𝒫 𝐵))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dssmapfvd.o | . . . 4 ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠)))))) | |
| 2 | dssmapfvd.d | . . . 4 ⊢ 𝐷 = (𝑂‘𝐵) | |
| 3 | dssmapfvd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
| 4 | 1, 2, 3 | dssmapnvod 44038 | . . 3 ⊢ (𝜑 → ◡𝐷 = 𝐷) | 
| 5 | 4 | coeq1d 5871 | . 2 ⊢ (𝜑 → (◡𝐷 ∘ 𝐷) = (𝐷 ∘ 𝐷)) | 
| 6 | 1, 2, 3 | dssmapf1od 44039 | . . 3 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵)) | 
| 7 | f1ococnv1 6876 | . . 3 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) → (◡𝐷 ∘ 𝐷) = ( I ↾ (𝒫 𝐵 ↑m 𝒫 𝐵))) | |
| 8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → (◡𝐷 ∘ 𝐷) = ( I ↾ (𝒫 𝐵 ↑m 𝒫 𝐵))) | 
| 9 | 5, 8 | eqtr3d 2778 | 1 ⊢ (𝜑 → (𝐷 ∘ 𝐷) = ( I ↾ (𝒫 𝐵 ↑m 𝒫 𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∖ cdif 3947 𝒫 cpw 4599 ↦ cmpt 5224 I cid 5576 ◡ccnv 5683 ↾ cres 5686 ∘ ccom 5688 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 ↑m cmap 8867 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-map 8869 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |