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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 44025 | . 2 ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑m 𝐴)⟶(𝒫 𝐴 ↑m 𝐵)) |
| 6 | fsovcnvlem.h | . . 3 ⊢ 𝐻 = (𝐵𝑂𝐴) | |
| 7 | 1, 3, 2, 6 | fsovfd 44025 | . 2 ⊢ (𝜑 → 𝐻:(𝒫 𝐴 ↑m 𝐵)⟶(𝒫 𝐵 ↑m 𝐴)) |
| 8 | 1, 2, 3, 4, 6 | fsovcnvlem 44026 | . 2 ⊢ (𝜑 → (𝐻 ∘ 𝐺) = ( I ↾ (𝒫 𝐵 ↑m 𝐴))) |
| 9 | 1, 3, 2, 6, 4 | fsovcnvlem 44026 | . 2 ⊢ (𝜑 → (𝐺 ∘ 𝐻) = ( I ↾ (𝒫 𝐴 ↑m 𝐵))) |
| 10 | 5, 7, 8, 9 | 2fcoidinvd 7315 | 1 ⊢ (𝜑 → ◡𝐺 = 𝐻) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 𝒫 cpw 4600 ↦ cmpt 5225 ◡ccnv 5684 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ↑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 ax-un 7755 |
| 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 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 |
| This theorem is referenced by: fsovcnvfvd 44028 fsovf1od 44029 ntrneicnv 44091 clsneicnv 44118 neicvgnvo 44128 neicvgel1 44132 |
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