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Theorem fnpm 6550
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 6545 . 2 pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓})
2 vex 2689 . . . . 5 𝑦 ∈ V
3 vex 2689 . . . . 5 𝑥 ∈ V
42, 3xpex 4654 . . . 4 (𝑦 × 𝑥) ∈ V
54pwex 4107 . . 3 𝒫 (𝑦 × 𝑥) ∈ V
65rabex 4072 . 2 {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓} ∈ V
71, 6fnmpoi 6102 1 pm Fn (V × V)
Colors of variables: wff set class
Syntax hints:  {crab 2420  Vcvv 2686  𝒫 cpw 3510   × cxp 4537  Fun wfun 5117   Fn wfn 5118  pm cpm 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pm 6545
This theorem is referenced by:  lmfval  12375
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