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

Theorem fnpm 8404
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 8399 . 2 pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
2 vex 3503 . . . . 5 𝑦 ∈ V
3 vex 3503 . . . . 5 𝑥 ∈ V
42, 3xpex 7465 . . . 4 (𝑦 × 𝑥) ∈ V
54pwex 5278 . . 3 𝒫 (𝑦 × 𝑥) ∈ V
65rabex 5232 . 2 {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} ∈ V
71, 6fnmpoi 7759 1 pm Fn (V × V)
Colors of variables: wff setvar class
Syntax hints:  {crab 3147  Vcvv 3500  𝒫 cpw 4542   × cxp 5552  Fun wfun 6346   Fn wfn 6347  pm cpm 8397
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-pm 8399
This theorem is referenced by:  elpmi  8415  pmresg  8424  pmsspw  8431
  Copyright terms: Public domain W3C validator