Theorem mapval2 8911

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 7119	. . . 4 ⊢ (𝑔:𝐵⟶𝐴 ↔ (𝑔 Fn 𝐵 ∧ 𝑔 ⊆ (𝐵 × 𝐴)))
2	1	biancomi 462	. . 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 8910	. . 3 ⊢ (𝑔 ∈ (𝐴 ↑m 𝐵) ↔ 𝑔:𝐵⟶𝐴)
6		elin 3979	. . . 4 ⊢ (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ∧ 𝑔 ∈ {𝑓 ∣ 𝑓 Fn 𝐵}))
7		velpw 4610	. . . . 5 ⊢ (𝑔 ∈ 𝒫 (𝐵 × 𝐴) ↔ 𝑔 ⊆ (𝐵 × 𝐴))
8		vex 3482	. . . . . 6 ⊢ 𝑔 ∈ V
9		fneq1 6660	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝐵 ↔ 𝑔 Fn 𝐵))
10	8, 9	elab 3681	. . . . 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 275	. . 3 ⊢ (𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) ↔ (𝑔 ⊆ (𝐵 × 𝐴) ∧ 𝑔 Fn 𝐵))
13	2, 5, 12	3bitr4i 303	. 2 ⊢ (𝑔 ∈ (𝐴 ↑m 𝐵) ↔ 𝑔 ∈ (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}))
14	13	eqriv 2732	1 ⊢ (𝐴 ↑m 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵})

Syntax hints:   ∧ wa 395   = wceq 1537   ∈ wcel 2106  {cab 2712  Vcvv 3478   ∩ cin 3962   ⊆ wss 3963  𝒫 cpw 4605   × cxp 5687   Fn wfn 6558  ⟶wf 6559  (class class class)co 7431   ↑_m cmap 8865

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867

This theorem is referenced by: (None)