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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsovd | Structured version Visualization version GIF version |
Description: Value of 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. (Contributed by RP, 25-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) |
Ref | Expression |
---|---|
fsovd | ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}))) |
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 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})))) |
3 | pweq 4538 | . . . . . 6 ⊢ (𝑏 = 𝐵 → 𝒫 𝑏 = 𝒫 𝐵) | |
4 | 3 | adantl 482 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝒫 𝑏 = 𝒫 𝐵) |
5 | simpl 483 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝑎 = 𝐴) | |
6 | 4, 5 | oveq12d 7163 | . . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝒫 𝑏 ↑m 𝑎) = (𝒫 𝐵 ↑m 𝐴)) |
7 | simpr 485 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵) | |
8 | rabeq 3481 | . . . . . 6 ⊢ (𝑎 = 𝐴 → {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) | |
9 | 8 | adantr 481 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}) |
10 | 7, 9 | mpteq12dv 5142 | . . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)}) = (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})) |
11 | 6, 10 | mpteq12dv 5142 | . . 3 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}))) |
12 | 11 | adantl 482 | . 2 ⊢ ((𝜑 ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → (𝑓 ∈ (𝒫 𝑏 ↑m 𝑎) ↦ (𝑦 ∈ 𝑏 ↦ {𝑥 ∈ 𝑎 ∣ 𝑦 ∈ (𝑓‘𝑥)})) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}))) |
13 | fsovd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
14 | 13 | elexd 3512 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
15 | fsovd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
16 | 15 | elexd 3512 | . 2 ⊢ (𝜑 → 𝐵 ∈ V) |
17 | ovex 7178 | . . . 4 ⊢ (𝒫 𝐵 ↑m 𝐴) ∈ V | |
18 | 17 | mptex 6977 | . . 3 ⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})) ∈ V |
19 | 18 | a1i 11 | . 2 ⊢ (𝜑 → (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)})) ∈ V) |
20 | 2, 12, 14, 16, 19 | ovmpod 7291 | 1 ⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ (𝑦 ∈ 𝐵 ↦ {𝑥 ∈ 𝐴 ∣ 𝑦 ∈ (𝑓‘𝑥)}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 Vcvv 3492 𝒫 cpw 4535 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 ↑m cmap 8395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 |
This theorem is referenced by: fsovrfovd 40233 fsovfvd 40234 fsovfd 40236 fsovcnvlem 40237 |
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