Theorem fnov 7536

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 6947	. 2 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)))
2		fveq2 6888	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3		df-ov 7408	. . . . 5 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
4	2, 3	eqtr4di 2790	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹‘𝑧) = (𝑥𝐹𝑦))
5	4	mpompt 7518	. . 3 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))
6	5	eqeq2i 2745	. 2 ⊢ (𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹‘𝑧)) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))
7	1, 6	bitri 274	1 ⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦)))

Syntax hints:   ↔ wb 205   = wceq 1541  ⟨cop 4633   ↦ cmpt 5230   × cxp 5673   Fn wfn 6535  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fn 6543  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410

This theorem is referenced by:  mapxpen  9139  dfioo2  13423  fnhomeqhomf  17631  reschomf  17775  cofulid  17836  cofurid  17837  prf1st  18152  prf2nd  18153  1st2ndprf  18154  curfuncf  18187  curf2ndf  18196  plusfeq  18565  scafeq  20484  cnfldsub  20965  ipfeq  21194  psrvscafval  21500  mdetunilem7  22111  madurid  22137  cnmpt22f  23170  cnmptcom  23173  xkocnv  23309  qustgplem  23616  stdbdxmet  24015  iimulcn  24445  rrxds  24901  rrxmfval  24914  cnnvm  29922  ofpreima  31877  ressplusf  32114  fedgmullem2  32703  matmpo  32771  mndpluscn  32894  rmulccn  32896  raddcn  32897  txsconnlem  34219  cvmlift2lem6  34287  cvmlift2lem7  34288  cvmlift2lem12  34293  unccur  36459  matunitlindflem1  36472  rngchomrnghmresALTV  46847  2arymaptfo  47293