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Theorem fnpm 7810
 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 7805 . 2 pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
2 vex 3189 . . . . 5 𝑦 ∈ V
3 vex 3189 . . . . 5 𝑥 ∈ V
42, 3xpex 6915 . . . 4 (𝑦 × 𝑥) ∈ V
54pwex 4808 . . 3 𝒫 (𝑦 × 𝑥) ∈ V
65rabex 4773 . 2 {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} ∈ V
71, 6fnmpt2i 7184 1 pm Fn (V × V)
 Colors of variables: wff setvar class Syntax hints:  {crab 2911  Vcvv 3186  𝒫 cpw 4130   × cxp 5072  Fun wfun 5841   Fn wfn 5842   ↑pm cpm 7803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-pm 7805 This theorem is referenced by:  elpmi  7820  pmresg  7829  pmsspw  7836
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