Theorem fnmap 8872

Description: Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)

Assertion

Ref	Expression
fnmap	⊢ ↑m Fn (V × V)

Proof of Theorem fnmap

Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-map 8867	. 2 ⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
2		mapex 7962	. . 3 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → {𝑓 ∣ 𝑓:𝑦⟶𝑥} ∈ V)
3	2	el2v 3485	. 2 ⊢ {𝑓 ∣ 𝑓:𝑦⟶𝑥} ∈ V
4	1, 3	fnmpoi 8094	1 ⊢ ↑m Fn (V × V)

Syntax hints:   ∈ wcel 2106  {cab 2712  Vcvv 3478   × cxp 5687   Fn wfn 6558  ⟶wf 6559   ↑_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-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  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-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867

This theorem is referenced by:  elmapex  8887