Theorem fnov 7531

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 6929	. 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)))
2		fveq2 6871	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3		df-ov 7403	. . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2818	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝑥𝐹𝑦))
5	4	mpompt 7514	. . 3 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))
6	5	eqeq2i 2778	. 2 ⊢ (𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
7	1, 6	bitri 278	1 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))

Syntax hints:   ↔ wb 209   = wceq 1563  ⟨cop 4591   ↦ cmpt 5186   × cxp 5650   Fn wfn 6520  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fn 6528  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405

This theorem is referenced by:  mapxpen  9119  dfioo2  13468  fnhomeqhomf  17737  reschomf  17878  cofulid  17937  cofurid  17938  prf1st  18250  prf2nd  18251  1st2ndprf  18252  curfuncf  18284  curf2ndf  18293  plusfeq  18696  scafeq  20972  cnfldadd  21488  cnfldmul  21490  cnfldsub  21510  ipfeq  21760  psrvscafval  22058  mdetunilem7  22736  madurid  22762  cnmpt22f  23793  cnmptcom  23796  xkocnv  23932  qustgplem  24239  stdbdxmet  24633  rrxds  25513  rrxmfval  25526  cnnvm  30943  ofpreima  32922  ressplusf  33196  elrgspnlem2  33476  fedgmullem2  33937  matmpo  34110  mndpluscn  34233  raddcn  34236  txsconnlem  35603  cvmlift2lem6  35671  cvmlift2lem7  35672  cvmlift2lem12  35677  unccur  38114  matunitlindflem1  38127  rngchomrnghmresALTV  48899  2arymaptfo  49285  isofval2  49661  funcf2lem2  49711  upeu4  49825  diag1  49933  fucofulem2  49940