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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 40356 | . . 3 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))))) |
5 | 3 | pwexd 5272 | . . . . . 6 ⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
6 | 5 | mptexd 6981 | . . . . 5 ⊢ (𝜑 → (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) ∈ V) |
7 | 6 | ralrimivw 3183 | . . . 4 ⊢ (𝜑 → ∀𝑓 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)(𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠)))) ∈ V) |
8 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑓(𝒫 𝐵 ↑m 𝒫 𝐵) | |
9 | 8 | fnmptf 6478 | . . . 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)) |
10 | 7, 9 | syl 17 | . . 3 ⊢ (𝜑 → (𝑓 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) ↦ (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝑓‘(𝐵 ∖ 𝑠))))) Fn (𝒫 𝐵 ↑m 𝒫 𝐵)) |
11 | fneq1 6438 | . . . 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))) | |
12 | 11 | biimprd 250 | . . 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))) |
13 | 4, 10, 12 | sylc 65 | . 2 ⊢ (𝜑 → 𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵)) |
14 | 1, 2, 3 | dssmapnvod 40359 | . 2 ⊢ (𝜑 → ◡𝐷 = 𝐷) |
15 | nvof1o 7031 | . 2 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ ◡𝐷 = 𝐷) → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵)) | |
16 | 13, 14, 15 | syl2anc 586 | 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 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ∖ cdif 3932 𝒫 cpw 4538 ↦ cmpt 5138 ◡ccnv 5548 Fn wfn 6344 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 |
This theorem is referenced by: dssmap2d 40361 ntrclsf1o 40394 clsneif1o 40447 clsneikex 40449 clsneinex 40450 clsneiel1 40451 neicvgf1o 40457 neicvgmex 40460 neicvgel1 40462 dssmapntrcls 40471 dssmapclsntr 40472 |
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