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Theorem restfn 16293
 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 16291 . 2 t = (𝑗 ∈ V, 𝑥 ∈ V ↦ ran (𝑦𝑗 ↦ (𝑦𝑥)))
2 vex 3354 . . . 4 𝑗 ∈ V
32mptex 6630 . . 3 (𝑦𝑗 ↦ (𝑦𝑥)) ∈ V
43rnex 7247 . 2 ran (𝑦𝑗 ↦ (𝑦𝑥)) ∈ V
51, 4fnmpt2i 7389 1 t Fn (V × V)
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3351   ∩ cin 3722   ↦ cmpt 4863   × cxp 5247  ran crn 5250   Fn wfn 6026   ↾t crest 16289 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-rest 16291 This theorem is referenced by:  0rest  16298  restsspw  16300  firest  16301  restrcl  21182  restbas  21183  ssrest  21201  resstopn  21211
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