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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 44031 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))))) |
| 8 | difeq2 4120 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝐵 ∖ 𝑠) = (𝐵 ∖ 𝑆)) | |
| 9 | 8 | fveq2d 6910 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝐹‘(𝐵 ∖ 𝑠)) = (𝐹‘(𝐵 ∖ 𝑆))) |
| 10 | 9 | difeq2d 4126 | . . . 4 ⊢ (𝑠 = 𝑆 → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
| 11 | 10 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑠 = 𝑆) → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
| 12 | dssmapfv3d.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
| 13 | 4 | difexd 5331 | . . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))) ∈ V) |
| 14 | 7, 11, 12, 13 | fvmptd 7023 | . 2 ⊢ (𝜑 → (𝐺‘𝑆) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
| 15 | 1, 14 | eqtrid 2789 | 1 ⊢ (𝜑 → 𝑇 = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 𝒫 cpw 4600 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: ntrclselnel1 44070 ntrclsfv 44072 ntrclscls00 44079 ntrclsiso 44080 ntrclsk2 44081 ntrclskb 44082 ntrclsk3 44083 ntrclsk13 44084 dssmapntrcls 44141 |
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