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Theorem restfn 17369
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 17367 . 2 t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦𝑗 ↦ (𝑦𝑥)))
2 vex 3470 . . . 4 𝑗 ∈ V
32mptex 7216 . . 3 (𝑦𝑗 ↦ (𝑦𝑥)) ∈ V
43rnex 7896 . 2 ran (𝑦𝑗 ↦ (𝑦𝑥)) ∈ V
51, 4fnmpoi 8049 1 t Fn (V × V)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3466  cin 3939  cmpt 5221   × cxp 5664  ran crn 5667   Fn wfn 6528  t crest 17365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-rest 17367
This theorem is referenced by:  0rest  17374  restsspw  17376  firest  17377  restrcl  22983  restbas  22984  ssrest  23002  resstopn  23012
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