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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 43590 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))))) |
8 | difeq2 4112 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝐵 ∖ 𝑠) = (𝐵 ∖ 𝑆)) | |
9 | 8 | fveq2d 6900 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝐹‘(𝐵 ∖ 𝑠)) = (𝐹‘(𝐵 ∖ 𝑆))) |
10 | 9 | difeq2d 4118 | . . . 4 ⊢ (𝑠 = 𝑆 → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
11 | 10 | adantl 480 | . . 3 ⊢ ((𝜑 ∧ 𝑠 = 𝑆) → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
12 | dssmapfv3d.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
13 | 4 | difexd 5332 | . . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆))) ∈ V) |
14 | 7, 11, 12, 13 | fvmptd 7011 | . 2 ⊢ (𝜑 → (𝐺‘𝑆) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
15 | 1, 14 | eqtrid 2777 | 1 ⊢ (𝜑 → 𝑇 = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ∖ cdif 3941 𝒫 cpw 4604 ↦ cmpt 5232 ‘cfv 6549 (class class class)co 7419 ↑m cmap 8845 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 |
This theorem is referenced by: ntrclselnel1 43629 ntrclsfv 43631 ntrclscls00 43638 ntrclsiso 43639 ntrclsk2 43640 ntrclskb 43641 ntrclsk3 43642 ntrclsk13 43643 dssmapntrcls 43700 |
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