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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 7097 | . . . 4 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 Fn 𝐵 ∧ 𝑔 ⊆ (𝐵 × 𝐴))) | |
2 | 1 | biancomi 463 | . . 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 8861 | . . 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 4606 | . . . . 5 ⊢ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑔 ⊆ (𝐵 × 𝐴)) | |
8 | vex 3478 | . . . . . 6 ⊢ 𝑔 ∈ V | |
9 | fneq1 6637 | . . . . . 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 627 | . . . 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 2729 | 1 ⊢ (𝐴 ↑m 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 {cab 2709 Vcvv 3474 ∩ cin 3946 ⊆ wss 3947 𝒫 cpw 4601 × cxp 5673 Fn wfn 6535 ⟶wf 6536 (class class class)co 7405 ↑m cmap 8816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-map 8818 |
This theorem is referenced by: (None) |
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