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Mirrors > Home > MPE Home > Th. List > mapval2 | Structured version Visualization version GIF version |
Description: Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.) |
Ref | Expression |
---|---|
elmap.1 | ⊢ 𝐴 ∈ V |
elmap.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
mapval2 | ⊢ (𝐴 ↑m 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff2 7112 | . . . 4 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 Fn 𝐵 ∧ 𝑔 ⊆ (𝐵 × 𝐴))) | |
2 | 1 | biancomi 461 | . . 3 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵)) |
3 | elmap.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | elmap.2 | . . . 4 ⊢ 𝐵 ∈ V | |
5 | 3, 4 | elmap 8894 | . . 3 ⊢ (𝑔 ∈ (𝐴 ↑m 𝐵) ↔ 𝑔:𝐵⟶𝐴) |
6 | elin 3963 | . . . 4 ⊢ (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵})) | |
7 | velpw 4609 | . . . . 5 ⊢ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑔 ⊆ (𝐵 × 𝐴)) | |
8 | vex 3475 | . . . . . 6 ⊢ 𝑔 ∈ V | |
9 | fneq1 6648 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐵 ↔ 𝑔 Fn 𝐵)) | |
10 | 8, 9 | elab 3667 | . . . . 5 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵} ↔ 𝑔 Fn 𝐵) |
11 | 7, 10 | anbi12i 626 | . . . 4 ⊢ ((𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵)) |
12 | 6, 11 | bitri 274 | . . 3 ⊢ (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵)) |
13 | 2, 5, 12 | 3bitr4i 302 | . 2 ⊢ (𝑔 ∈ (𝐴 ↑m 𝐵) ↔ 𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵})) |
14 | 13 | eqriv 2724 | 1 ⊢ (𝐴 ↑m 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∈ wcel 2098 {cab 2704 Vcvv 3471 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4604 × cxp 5678 Fn wfn 6546 ⟶wf 6547 (class class class)co 7424 ↑m cmap 8849 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
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 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-map 8851 |
This theorem is referenced by: (None) |
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