Theorem mapval2 6732

Description: Alternate expression for the value of set exponentiation. (Contributed by NM, 3-Nov-2007.)

Hypotheses

Ref	Expression
elmap.1	⊢ 𝐴 ∈ V
elmap.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
mapval2	⊢ (𝐴 ↑𝑚 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵})

Distinct variable group:   𝐵,𝑓

Allowed substitution hint:   𝐴(𝑓)

Proof of Theorem mapval2

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dff2 5702	. . . 4 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 Fn 𝐵 ∧ 𝑔 ⊆ (𝐵 × 𝐴)))
2		ancom 266	. . . 4 ⊢ ((𝑔 Fn 𝐵 ∧ 𝑔 ⊆ (𝐵 × 𝐴)) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
3	1, 2	bitri 184	. . 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 6731	. . 3 ⊢ (𝑔 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝑔:𝐵⟶𝐴)
7		elin 3342	. . . 4 ⊢ (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵}))
8		velpw 3608	. . . . 5 ⊢ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑔 ⊆ (𝐵 × 𝐴))
9		vex 2763	. . . . . 6 ⊢ 𝑔 ∈ V
10		fneq1 5342	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐵 ↔ 𝑔 Fn 𝐵))
11	9, 10	elab 2904	. . . . 5 ⊢ (𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵} ↔ 𝑔 Fn 𝐵)
12	8, 11	anbi12i 460	. . . 4 ⊢ ((𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
13	7, 12	bitri 184	. . 3 ⊢ (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
14	3, 6, 13	3bitr4i 212	. 2 ⊢ (𝑔 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}))
15	14	eqriv 2190	1 ⊢ (𝐴 ↑𝑚 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵})

Syntax hints:   ∧ wa 104   = wceq 1364   ∈ wcel 2164  {cab 2179  Vcvv 2760   ∩ cin 3152   ⊆ wss 3153  𝒫 cpw 3601   × cxp 4657   Fn wfn 5249  ⟶wf 5250  (class class class)co 5918   ↑_𝑚 cmap 6702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-map 6704

This theorem is referenced by: (None)