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Theorem fnpm 8875
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 8870 . 2 pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
2 vex 3483 . . . . 5 𝑦 ∈ V
3 vex 3483 . . . . 5 𝑥 ∈ V
42, 3xpex 7774 . . . 4 (𝑦 × 𝑥) ∈ V
54pwex 5379 . . 3 𝒫 (𝑦 × 𝑥) ∈ V
65rabex 5338 . 2 {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} ∈ V
71, 6fnmpoi 8096 1 pm Fn (V × V)
Colors of variables: wff setvar class
Syntax hints:  {crab 3435  Vcvv 3479  𝒫 cpw 4599   × cxp 5682  Fun wfun 6554   Fn wfn 6555  pm cpm 8868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-pm 8870
This theorem is referenced by:  elpmi  8887  pmresg  8911  pmsspw  8918
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