MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnpm Structured version   Visualization version   GIF version

Theorem fnpm 8863
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.
StepHypRef Expression
1 df-pm 8858 . 2 pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
2 vex 3466 . . . . 5 𝑦 ∈ V
3 vex 3466 . . . . 5 𝑥 ∈ V
42, 3xpex 7761 . . . 4 (𝑦 × 𝑥) ∈ V
54pwex 5384 . . 3 𝒫 (𝑦 × 𝑥) ∈ V
65rabex 5339 . 2 {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} ∈ V
71, 6fnmpoi 8084 1 pm Fn (V × V)
Colors of variables: wff setvar class
Syntax hints:  {crab 3419  Vcvv 3462  𝒫 cpw 4607   × cxp 5680  Fun wfun 6548   Fn wfn 6549  pm cpm 8856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-pm 8858
This theorem is referenced by:  elpmi  8875  pmresg  8899  pmsspw  8906
  Copyright terms: Public domain W3C validator