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Theorem fnov 6721
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 6198 . 2 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)))
2 fveq2 6148 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 6607 . . . . 5 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3syl6eqr 2673 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝑥𝐹𝑦))
54mpt2mpt 6705 . . 3 (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)) = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦))
65eqeq2i 2633 . 2 (𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
71, 6bitri 264 1 (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  cop 4154  cmpt 4673   × cxp 5072   Fn wfn 5842  cfv 5847  (class class class)co 6604  cmpt2 6606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609
This theorem is referenced by:  mapxpen  8070  dfioo2  12216  fnhomeqhomf  16272  reschomf  16412  cofulid  16471  cofurid  16472  prf1st  16765  prf2nd  16766  1st2ndprf  16767  curfuncf  16799  curf2ndf  16808  plusfeq  17170  scafeq  18804  psrvscafval  19309  cnfldsub  19693  ipfeq  19914  mdetunilem7  20343  madurid  20369  cnmpt22f  21388  cnmptcom  21391  xkocnv  21527  qustgplem  21834  stdbdxmet  22230  iimulcn  22645  rrxds  23089  rrxmfval  23097  cnnvm  27383  ofpreima  29305  ressplusf  29432  matmpt2  29648  mndpluscn  29751  rmulccn  29753  raddcn  29754  txsconnlem  30927  cvmlift2lem6  30995  cvmlift2lem7  30996  cvmlift2lem12  31001  unccur  33021  matunitlindflem1  33034  rngchomrnghmresALTV  41281
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