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Mirrors > Home > ILE Home > Th. List > mapval2 | 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 5610 | . . . 4 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 Fn 𝐵 ∧ 𝑔 ⊆ (𝐵 × 𝐴))) | |
2 | ancom 264 | . . . 4 ⊢ ((𝑔 Fn 𝐵 ∧ 𝑔 ⊆ (𝐵 × 𝐴)) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵)) | |
3 | 1, 2 | bitri 183 | . . 3 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵)) |
4 | elmap.1 | . . . 4 ⊢ 𝐴 ∈ V | |
5 | elmap.2 | . . . 4 ⊢ 𝐵 ∈ V | |
6 | 4, 5 | elmap 6619 | . . 3 ⊢ (𝑔 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶𝐴) |
7 | elin 3290 | . . . 4 ⊢ (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵})) | |
8 | velpw 3550 | . . . . 5 ⊢ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑔 ⊆ (𝐵 × 𝐴)) | |
9 | vex 2715 | . . . . . 6 ⊢ 𝑔 ∈ V | |
10 | fneq1 5257 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐵 ↔ 𝑔 Fn 𝐵)) | |
11 | 9, 10 | elab 2856 | . . . . 5 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵} ↔ 𝑔 Fn 𝐵) |
12 | 8, 11 | anbi12i 456 | . . . 4 ⊢ ((𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵)) |
13 | 7, 12 | bitri 183 | . . 3 ⊢ (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵)) |
14 | 3, 6, 13 | 3bitr4i 211 | . 2 ⊢ (𝑔 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵})) |
15 | 14 | eqriv 2154 | 1 ⊢ (𝐴 ↑𝑚 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1335 ∈ wcel 2128 {cab 2143 Vcvv 2712 ∩ cin 3101 ⊆ wss 3102 𝒫 cpw 3543 × cxp 4583 Fn wfn 5164 ⟶wf 5165 (class class class)co 5821 ↑𝑚 cmap 6590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-fv 5177 df-ov 5824 df-oprab 5825 df-mpo 5826 df-map 6592 |
This theorem is referenced by: (None) |
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