Theorem fnpm 8581

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 8576	. 2 ⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
2		vex 3426	. . . . 5 ⊢ 𝑦 ∈ V
3		vex 3426	. . . . 5 ⊢ 𝑥 ∈ V
4	2, 3	xpex 7581	. . . 4 ⊢ (𝑦 × 𝑥) ∈ V
5	4	pwex 5298	. . 3 ⊢ 𝒫 (𝑦 × 𝑥) ∈ V
6	5	rabex 5251	. 2 ⊢ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} ∈ V
7	1, 6	fnmpoi 7883	1 ⊢ ↑pm Fn (V × V)

Syntax hints:  {crab 3067  Vcvv 3422  𝒫 cpw 4530   × cxp 5578  Fun wfun 6412   Fn wfn 6413   ↑_pm cpm 8574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-pm 8576

This theorem is referenced by:  elpmi  8592  pmresg  8616  pmsspw  8623