Theorem fnov 7477

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 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (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.

Step	Hyp	Ref	Expression
1		dffn5 6880	. 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)))
2		fveq2 6822	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3		df-ov 7349	. . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2784	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝑥𝐹𝑦))
5	4	mpompt 7460	. . 3 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))
6	5	eqeq2i 2744	. 2 ⊢ (𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
7	1, 6	bitri 275	1 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))

Syntax hints:   ↔ wb 206   = wceq 1541  ⟨cop 4582   ↦ cmpt 5172   × cxp 5614   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351

This theorem is referenced by:  mapxpen  9056  dfioo2  13347  fnhomeqhomf  17594  reschomf  17735  cofulid  17794  cofurid  17795  prf1st  18107  prf2nd  18108  1st2ndprf  18109  curfuncf  18141  curf2ndf  18150  plusfeq  18553  scafeq  20813  cnfldadd  21295  cnfldmul  21297  dfcnfldOLD  21305  cnfldsub  21332  ipfeq  21585  psrvscafval  21883  mdetunilem7  22531  madurid  22557  cnmpt22f  23588  cnmptcom  23591  xkocnv  23727  qustgplem  24034  stdbdxmet  24428  iimulcnOLD  24860  rrxds  25318  rrxmfval  25331  cnnvm  30657  ofpreima  32642  ressplusf  32939  elrgspnlem2  33205  fedgmullem2  33638  matmpo  33811  mndpluscn  33934  raddcn  33937  txsconnlem  35272  cvmlift2lem6  35340  cvmlift2lem7  35341  cvmlift2lem12  35346  unccur  37642  matunitlindflem1  37655  rngchomrnghmresALTV  48309  2arymaptfo  48685  isofval2  49063  funcf2lem2  49113  upeu4  49227  diag1  49335  fucofulem2  49342