Theorem fnmap 8873

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 8868	. 2 ⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥})
2		mapex 7963	. . 3 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → {𝑓 ∣ 𝑓:𝑦⟶𝑥} ∈ V)
3	2	el2v 3487	. 2 ⊢ {𝑓 ∣ 𝑓:𝑦⟶𝑥} ∈ V
4	1, 3	fnmpoi 8095	1 ⊢ ↑m Fn (V × V)

Syntax hints:   ∈ wcel 2108  {cab 2714  Vcvv 3480   × cxp 5683   Fn wfn 6556  ⟶wf 6557   ↑_m cmap 8866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868

This theorem is referenced by:  elmapex  8888