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Theorem fnov 7488
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 6902 . 2 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)))
2 fveq2 6843 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 7361 . . . . 5 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3eqtr4di 2791 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝑥𝐹𝑦))
54mpompt 7471 . . 3 (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)) = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦))
65eqeq2i 2746 . 2 (𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
71, 6bitri 275 1 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  cop 4593  cmpt 5189   × cxp 5632   Fn wfn 6492  cfv 6497  (class class class)co 7358  cmpo 7360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363
This theorem is referenced by:  mapxpen  9090  dfioo2  13373  fnhomeqhomf  17576  reschomf  17720  cofulid  17781  cofurid  17782  prf1st  18097  prf2nd  18098  1st2ndprf  18099  curfuncf  18132  curf2ndf  18141  plusfeq  18510  scafeq  20357  cnfldsub  20841  ipfeq  21070  psrvscafval  21374  mdetunilem7  21983  madurid  22009  cnmpt22f  23042  cnmptcom  23045  xkocnv  23181  qustgplem  23488  stdbdxmet  23887  iimulcn  24317  rrxds  24773  rrxmfval  24786  cnnvm  29666  ofpreima  31627  ressplusf  31866  fedgmullem2  32382  matmpo  32441  mndpluscn  32564  rmulccn  32566  raddcn  32567  txsconnlem  33891  cvmlift2lem6  33959  cvmlift2lem7  33960  cvmlift2lem12  33965  unccur  36107  matunitlindflem1  36120  rngchomrnghmresALTV  46380  2arymaptfo  46826
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