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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsovcnvfvd | Structured version Visualization version GIF version |
Description: The value of the converse of (𝐴𝑂𝐵), 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, evaluated at function 𝐹. (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) |
fsovcnvfvd.f | ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐴 ↑m 𝐵)) |
Ref | Expression |
---|---|
fsovcnvfvd | ⊢ (𝜑 → (◡𝐺‘𝐹) = (𝑦 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ 𝑦 ∈ (𝐹‘𝑥)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsovd.fs | . . . 4 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)}))) | |
2 | fsovd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | fsovd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
4 | fsovfvd.g | . . . 4 ⊢ 𝐺 = (𝐴𝑂𝐵) | |
5 | eqid 2740 | . . . 4 ⊢ (𝐵𝑂𝐴) = (𝐵𝑂𝐴) | |
6 | 1, 2, 3, 4, 5 | fsovcnvd 41592 | . . 3 ⊢ (𝜑 → ◡𝐺 = (𝐵𝑂𝐴)) |
7 | 6 | fveq1d 6773 | . 2 ⊢ (𝜑 → (◡𝐺‘𝐹) = ((𝐵𝑂𝐴)‘𝐹)) |
8 | fsovcnvfvd.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝒫 𝐴 ↑m 𝐵)) | |
9 | 1, 3, 2, 5, 8 | fsovfvd 41588 | . 2 ⊢ (𝜑 → ((𝐵𝑂𝐴)‘𝐹) = (𝑦 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ 𝑦 ∈ (𝐹‘𝑥)})) |
10 | 7, 9 | eqtrd 2780 | 1 ⊢ (𝜑 → (◡𝐺‘𝐹) = (𝑦 ∈ 𝐴 ↦ {𝑥 ∈ 𝐵 ∣ 𝑦 ∈ (𝐹‘𝑥)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 {crab 3070 Vcvv 3431 𝒫 cpw 4539 ↦ cmpt 5162 ◡ccnv 5589 ‘cfv 6432 (class class class)co 7271 ∈ cmpo 7273 ↑m cmap 8598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-oprab 7275 df-mpo 7276 df-1st 7824 df-2nd 7825 df-map 8600 |
This theorem is referenced by: (None) |
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