Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6860 | . . . 4 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 Fn 𝐵 ∧ 𝑔 ⊆ (𝐵 × 𝐴))) | |
2 | 1 | biancomi 465 | . . 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 8429 | . . 3 ⊢ (𝑔 ∈ (𝐴 ↑m 𝐵) ↔ 𝑔:𝐵⟶𝐴) |
6 | elin 4169 | . . . 4 ⊢ (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵})) | |
7 | velpw 4547 | . . . . 5 ⊢ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑔 ⊆ (𝐵 × 𝐴)) | |
8 | vex 3498 | . . . . . 6 ⊢ 𝑔 ∈ V | |
9 | fneq1 6439 | . . . . . 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 628 | . . . 4 ⊢ ((𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵)) |
12 | 6, 11 | bitri 277 | . . 3 ⊢ (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵)) |
13 | 2, 5, 12 | 3bitr4i 305 | . 2 ⊢ (𝑔 ∈ (𝐴 ↑m 𝐵) ↔ 𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵})) |
14 | 13 | eqriv 2818 | 1 ⊢ (𝐴 ↑m 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 {cab 2799 Vcvv 3495 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 × cxp 5548 Fn wfn 6345 ⟶wf 6346 (class class class)co 7150 ↑m cmap 8400 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |