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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 37332 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑠 ∈ 𝒫 𝐵 ↦ (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑠))))) |
8 | difeq2 3684 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝐵 ∖ 𝑠) = (𝐵 ∖ 𝑆)) | |
9 | 8 | fveq2d 6107 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝐹‘(𝐵 ∖ 𝑠)) = (𝐹‘(𝐵 ∖ 𝑆))) |
10 | 9 | difeq2d 3690 | . . . 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 | difexg 4735 | . . . 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 6197 | . 2 ⊢ (𝜑 → (𝐺‘𝑆) = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
16 | 1, 15 | syl5eq 2656 | 1 ⊢ (𝜑 → 𝑇 = (𝐵 ∖ (𝐹‘(𝐵 ∖ 𝑆)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 𝒫 cpw 4108 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 |
This theorem is referenced by: ntrclselnel1 37375 ntrclsfv 37377 ntrclscls00 37384 ntrclsiso 37385 ntrclsk2 37386 ntrclskb 37387 ntrclsk3 37388 ntrclsk13 37389 dssmapntrcls 37446 |
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