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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsovcnvd | Structured version Visualization version GIF version |
Description: The value of the converse (𝐴𝑂𝐵) is (𝐵𝑂𝐴), where 𝑂 is the operator which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, gives a family of functions that include their own inverse. (Contributed by RP, 27-Apr-2021.) |
Ref | Expression |
---|---|
fsovd.fs | ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)}))) |
fsovd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
fsovd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
fsovfvd.g | ⊢ 𝐺 = (𝐴𝑂𝐵) |
fsovcnvlem.h | ⊢ 𝐻 = (𝐵𝑂𝐴) |
Ref | Expression |
---|---|
fsovcnvd | ⊢ (𝜑 → ◡𝐺 = 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsovd.fs | . . 3 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)}))) | |
2 | fsovd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | fsovd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | fsovfvd.g | . . 3 ⊢ 𝐺 = (𝐴𝑂𝐵) | |
5 | 1, 2, 3, 4 | fsovfd 43473 | . 2 ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑m 𝐴)⟶(𝒫 𝐴 ↑m 𝐵)) |
6 | fsovcnvlem.h | . . 3 ⊢ 𝐻 = (𝐵𝑂𝐴) | |
7 | 1, 3, 2, 6 | fsovfd 43473 | . 2 ⊢ (𝜑 → 𝐻:(𝒫 𝐴 ↑m 𝐵)⟶(𝒫 𝐵 ↑m 𝐴)) |
8 | 1, 2, 3, 4, 6 | fsovcnvlem 43474 | . 2 ⊢ (𝜑 → (𝐻 ∘ 𝐺) = ( I ↾ (𝒫 𝐵 ↑m 𝐴))) |
9 | 1, 3, 2, 6, 4 | fsovcnvlem 43474 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝐻) = ( I ↾ (𝒫 𝐴 ↑m 𝐵))) |
10 | 5, 7, 8, 9 | 2fcoidinvd 7310 | 1 ⊢ (𝜑 → ◡𝐺 = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3430 Vcvv 3473 𝒫 cpw 4606 ↦ cmpt 5235 ◡ccnv 5681 ‘cfv 6553 (class class class)co 7426 ∈ cmpo 7428 ↑m cmap 8851 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-map 8853 |
This theorem is referenced by: fsovcnvfvd 43476 fsovf1od 43477 ntrneicnv 43539 clsneicnv 43566 neicvgnvo 43576 neicvgel1 43580 |
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