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Mirrors > Home > MPE Home > Th. List > Mathboxes > dssmapf1od | Structured version Visualization version GIF version |
Description: For any base set 𝐵 the duality operator for self-mappings of subsets of that base set is one-to-one and onto. (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 |
---|---|
dssmapf1od | ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(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 | dssmapfvd 39089 | . . 3 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))))) |
5 | 3 | pwexd 5049 | . . . . . 6 ⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
6 | mptexg 6713 | . . . . . 6 ⊢ (𝒫 𝐵 ∈ V → (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) ∈ V) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) ∈ V) |
8 | 7 | ralrimivw 3148 | . . . 4 ⊢ (𝜑 → ∀𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)(𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) ∈ V) |
9 | nfcv 2941 | . . . . 5 ⊢ Ⅎ𝑓(𝒫 𝐵 ↑𝑚 𝒫 𝐵) | |
10 | 9 | fnmptf 6227 | . . . 4 ⊢ (∀𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)(𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) ∈ V → (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
11 | 8, 10 | syl 17 | . . 3 ⊢ (𝜑 → (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
12 | fneq1 6190 | . . . 4 ⊢ (𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) → (𝐷 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↔ (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵))) | |
13 | 12 | biimprd 240 | . . 3 ⊢ (𝐷 = (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))))) Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) → 𝐷 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵))) |
14 | 4, 11, 13 | sylc 65 | . 2 ⊢ (𝜑 → 𝐷 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
15 | 1, 2, 3 | dssmapnvod 39092 | . 2 ⊢ (𝜑 → ◡𝐷 = 𝐷) |
16 | nvof1o 6764 | . 2 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑𝑚 𝒫 𝐵) ∧ ◡𝐷 = 𝐷) → 𝐷:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵)) | |
17 | 14, 15, 16 | syl2anc 580 | 1 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ∀wral 3089 Vcvv 3385 ∖ cdif 3766 𝒫 cpw 4349 ↦ cmpt 4922 ◡ccnv 5311 Fn wfn 6096 –1-1-onto→wf1o 6100 ‘cfv 6101 (class class class)co 6878 ↑𝑚 cmap 8095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-1st 7401 df-2nd 7402 df-map 8097 |
This theorem is referenced by: dssmap2d 39094 ntrclsf1o 39127 clsneif1o 39180 clsneikex 39182 clsneinex 39183 clsneiel1 39184 neicvgf1o 39190 neicvgmex 39193 neicvgel1 39195 dssmapntrcls 39204 dssmapclsntr 39205 |
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