![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fsovfd | Structured version Visualization version GIF version |
Description: The operator, (𝐴𝑂𝐵), 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, 𝐴 and 𝐵, gives a function between two sets of functions. (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) |
Ref | Expression |
---|---|
fsovfd | ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑m 𝐴)⟶(𝒫 𝐴 ↑m 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsovfvd.g | . . 3 ⊢ 𝐺 = (𝐴𝑂𝐵) | |
2 | fsovd.fs | . . . 4 ⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)}))) | |
3 | fsovd.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | fsovd.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
5 | 2, 3, 4 | fsovd 43998 | . . 3 ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}))) |
6 | 1, 5 | eqtrid 2787 | . 2 ⊢ (𝜑 → 𝐺 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}))) |
7 | ssrab2 4090 | . . . . . . . 8 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} ⊆ 𝐴 | |
8 | 7 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} ⊆ 𝐴) |
9 | 3, 8 | sselpwd 5334 | . . . . . 6 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} ∈ 𝒫 𝐴) |
10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)} ∈ 𝒫 𝐴) |
11 | 10 | fmpttd 7135 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}):𝐵⟶𝒫 𝐴) |
12 | 3 | pwexd 5385 | . . . . 5 ⊢ (𝜑 → 𝒫 𝐴 ∈ V) |
13 | 12, 4 | elmapd 8879 | . . . 4 ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) ∈ (𝒫 𝐴 ↑m 𝐵) ↔ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}):𝐵⟶𝒫 𝐴)) |
14 | 11, 13 | mpbird 257 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) ∈ (𝒫 𝐴 ↑m 𝐵)) |
15 | 14 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) ∈ (𝒫 𝐴 ↑m 𝐵)) |
16 | 6, 15 | fmpt3d 7136 | 1 ⊢ (𝜑 → 𝐺:(𝒫 𝐵 ↑m 𝐴)⟶(𝒫 𝐴 ↑m 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ⊆ wss 3963 𝒫 cpw 4605 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ↑m cmap 8865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 |
This theorem is referenced by: fsovcnvd 44004 fsovf1od 44006 clsneiel1 44098 neicvgmex 44107 neicvgel1 44109 |
Copyright terms: Public domain | W3C validator |