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Theorem restfn 13329
Description: The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
Assertion
Ref Expression
restfn  |-t  Fn  ( _V  X.  _V )

Proof of Theorem restfn
Dummy variables  x  j  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rest 13327 . 2  |-t  =  ( j  e.  _V ,  x  e. 
_V  |->  ran  ( y  e.  j  |->  ( y  i^i  x ) ) )
2 vex 2791 . . . 4  |-  j  e. 
_V
32mptex 5746 . . 3  |-  ( y  e.  j  |->  ( y  i^i  x ) )  e.  _V
43rnex 4942 . 2  |-  ran  (
y  e.  j  |->  ( y  i^i  x ) )  e.  _V
51, 4fnmpt2i 6193 1  |-t  Fn  ( _V  X.  _V )
Colors of variables: wff set class
Syntax hints:   _Vcvv 2788    i^i cin 3151    e. cmpt 4077    X. cxp 4687   ran crn 4690    Fn wfn 5250   ↾t crest 13325
This theorem is referenced by:  0rest  13334  restsspw  13336  firest  13337  restrcl  16888  restbas  16889  ssrest  16907  resstopn  16916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-rest 13327
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