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Mirrors > Home > MPE Home > Th. List > Mathboxes > dssmapfv3d | Structured version Visualization version GIF version |
Description: Value of the duality operator for self-mappings of subsets of a base set, 𝐵 when applied to function 𝐹 and subset 𝑆. (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) |
dssmapfv2d.f | ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
dssmapfv2d.g | ⊢ 𝐺 = (𝐷‘𝐹) |
dssmapfv3d.s | ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) |
dssmapfv3d.t | ⊢ 𝑇 = (𝐺‘𝑆) |
Ref | Expression |
---|---|
dssmapfv3d | ⊢ (𝜑 → 𝑇 = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dssmapfv3d.t | . 2 ⊢ 𝑇 = (𝐺‘𝑆) | |
2 | dssmapfvd.o | . . . 4 ⊢ 𝑂 = (𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑𝑚 𝒫 𝑏) ↦ (𝑠 ∈ 𝒫 𝑏 ↦ (𝑏 ∖ (𝑓‘(𝑏 ∖ 𝑠)))))) | |
3 | dssmapfvd.d | . . . 4 ⊢ 𝐷 = (𝑂‘𝐵) | |
4 | dssmapfvd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
5 | dssmapfv2d.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) | |
6 | dssmapfv2d.g | . . . 4 ⊢ 𝐺 = (𝐷‘𝐹) | |
7 | 2, 3, 4, 5, 6 | dssmapfv2d 39082 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))))) |
8 | difeq2 3918 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝐵 ∖ 𝑠) = (𝐵 ∖ 𝑆)) | |
9 | 8 | fveq2d 6413 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝐹‘(𝐵 ∖ 𝑠)) = (𝐹‘(𝐵 ∖ 𝑆))) |
10 | 9 | difeq2d 3924 | . . . 4 ⊢ (𝑠 = 𝑆 → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
11 | 10 | adantl 474 | . . 3 ⊢ ((𝜑 ∧ 𝑠 = 𝑆) → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
12 | dssmapfv3d.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
13 | difexg 5001 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))) ∈ V) | |
14 | 4, 13 | syl 17 | . . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))) ∈ V) |
15 | 7, 11, 12, 14 | fvmptd 6511 | . 2 ⊢ (𝜑 → (𝐺‘𝑆) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
16 | 1, 15 | syl5eq 2843 | 1 ⊢ (𝜑 → 𝑇 = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 Vcvv 3383 ∖ cdif 3764 𝒫 cpw 4347 ↦ cmpt 4920 ‘cfv 6099 (class class class)co 6876 ↑𝑚 cmap 8093 |
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-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 |
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 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-ral 3092 df-rex 3093 df-reu 3094 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-ov 6879 |
This theorem is referenced by: ntrclselnel1 39125 ntrclsfv 39127 ntrclscls00 39134 ntrclsiso 39135 ntrclsk2 39136 ntrclskb 39137 ntrclsk3 39138 ntrclsk13 39139 dssmapntrcls 39196 |
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