MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnov Structured version   Visualization version   GIF version

Theorem fnov 7275
Description: Representation of a function in terms of its values. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
fnov (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem fnov
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dffn5 6715 . 2 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)))
2 fveq2 6661 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 7152 . . . . 5 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3syl6eqr 2877 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝑥𝐹𝑦))
54mpompt 7259 . . 3 (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)) = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦))
65eqeq2i 2837 . 2 (𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
71, 6bitri 278 1 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  cop 4556  cmpt 5132   × cxp 5540   Fn wfn 6338  cfv 6343  (class class class)co 7149  cmpo 7151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154
This theorem is referenced by:  mapxpen  8680  dfioo2  12837  fnhomeqhomf  16961  reschomf  17101  cofulid  17160  cofurid  17161  prf1st  17454  prf2nd  17455  1st2ndprf  17456  curfuncf  17488  curf2ndf  17497  plusfeq  17860  scafeq  19654  psrvscafval  20170  cnfldsub  20573  ipfeq  20794  mdetunilem7  21227  madurid  21253  cnmpt22f  22283  cnmptcom  22286  xkocnv  22422  qustgplem  22729  stdbdxmet  23125  iimulcn  23546  rrxds  24000  rrxmfval  24013  cnnvm  28468  ofpreima  30421  ressplusf  30648  fedgmullem2  31086  matmpo  31128  mndpluscn  31226  rmulccn  31228  raddcn  31229  txsconnlem  32544  cvmlift2lem6  32612  cvmlift2lem7  32613  cvmlift2lem12  32618  unccur  34985  matunitlindflem1  34998  rngchomrnghmresALTV  44546  2arymaptfo  44994
  Copyright terms: Public domain W3C validator