Theorem fnpm 8825

Description: Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.)

Assertion

Ref	Expression
fnpm	⊢ ↑pm Fn (V × V)

Proof of Theorem fnpm

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

Step	Hyp	Ref	Expression
1		df-pm 8820	. 2 ⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
2		vex 3470	. . . . 5 ⊢ 𝑦 ∈ V
3		vex 3470	. . . . 5 ⊢ 𝑥 ∈ V
4	2, 3	xpex 7734	. . . 4 ⊢ (𝑦 × 𝑥) ∈ V
5	4	pwex 5369	. . 3 ⊢ 𝒫 (𝑦 × 𝑥) ∈ V
6	5	rabex 5323	. 2 ⊢ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} ∈ V
7	1, 6	fnmpoi 8050	1 ⊢ ↑pm Fn (V × V)

Syntax hints:  {crab 3424  Vcvv 3466  𝒫 cpw 4595   × cxp 5665  Fun wfun 6528   Fn wfn 6529   ↑_pm cpm 8818

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-pm 8820

This theorem is referenced by:  elpmi  8837  pmresg  8861  pmsspw  8868