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Theorem restfn 16557
Description: The subspace topology operator is a function on pairs. (Contributed by Mario Carneiro, 1-May-2015.)
Assertion
Ref Expression
restfn t Fn (V × V)

Proof of Theorem restfn
Dummy variables 𝑥 𝑗 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rest 16555 . 2 t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦𝑗 ↦ (𝑦𝑥)))
2 vex 3418 . . . 4 𝑗 ∈ V
32mptex 6814 . . 3 (𝑦𝑗 ↦ (𝑦𝑥)) ∈ V
43rnex 7434 . 2 ran (𝑦𝑗 ↦ (𝑦𝑥)) ∈ V
51, 4fnmpoi 7578 1 t Fn (V × V)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3415  cin 3830  cmpt 5009   × cxp 5406  ran crn 5409   Fn wfn 6185  t crest 16553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-oprab 6982  df-mpo 6983  df-1st 7503  df-2nd 7504  df-rest 16555
This theorem is referenced by:  0rest  16562  restsspw  16564  firest  16565  restrcl  21472  restbas  21473  ssrest  21491  resstopn  21501
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