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Theorem fnov 7271
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 6717 . 2 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)))
2 fveq2 6663 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 7148 . . . . 5 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3syl6eqr 2871 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝑥𝐹𝑦))
54mpompt 7255 . . 3 (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)) = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦))
65eqeq2i 2831 . 2 (𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
71, 6bitri 276 1 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  cop 4563  cmpt 5137   × cxp 5546   Fn wfn 6343  cfv 6348  (class class class)co 7145  cmpo 7147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150
This theorem is referenced by:  mapxpen  8671  dfioo2  12826  fnhomeqhomf  16949  reschomf  17089  cofulid  17148  cofurid  17149  prf1st  17442  prf2nd  17443  1st2ndprf  17444  curfuncf  17476  curf2ndf  17485  plusfeq  17848  scafeq  19583  psrvscafval  20098  cnfldsub  20501  ipfeq  20722  mdetunilem7  21155  madurid  21181  cnmpt22f  22211  cnmptcom  22214  xkocnv  22350  qustgplem  22656  stdbdxmet  23052  iimulcn  23469  rrxds  23923  rrxmfval  23936  cnnvm  28386  ofpreima  30338  ressplusf  30564  fedgmullem2  30925  matmpo  30967  mndpluscn  31068  rmulccn  31070  raddcn  31071  txsconnlem  32384  cvmlift2lem6  32452  cvmlift2lem7  32453  cvmlift2lem12  32458  unccur  34756  matunitlindflem1  34769  rngchomrnghmresALTV  44195
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