Theorem fnov 6768

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 X. B ) <-> F = ( x e. A , y e. B |-> ( x F y ) ) )

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 6241	. 2 |- ( F Fn ( A X. B ) <-> F = ( z e. ( A X. B ) |-> ( F ` z ) ) )
2		fveq2 6191	. . . . 5 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		df-ov 6653	. . . . 5 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	syl6eqr 2674	. . . 4 |- ( z = <. x , y >. -> ( F ` z ) = ( x F y ) )
5	4	mpt2mpt 6752	. . 3 |- ( z e. ( A X. B ) |-> ( F ` z ) ) = ( x e. A , y e. B |-> ( x F y ) )
6	5	eqeq2i 2634	. 2 |- ( F = ( z e. ( A X. B ) |-> ( F ` z ) ) <-> F = ( x e. A , y e. B |-> ( x F y ) ) )
7	1, 6	bitri 264	1 |- ( F Fn ( A X. B ) <-> F = ( x e. A , y e. B |-> ( x F y ) ) )

Syntax hints:   <-> wb 196   = wceq 1483   <.cop 4183   |-> cmpt 4729   X. cxp 5112   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  mapxpen  8126  dfioo2  12274  fnhomeqhomf  16351  reschomf  16491  cofulid  16550  cofurid  16551  prf1st  16844  prf2nd  16845  1st2ndprf  16846  curfuncf  16878  curf2ndf  16887  plusfeq  17249  scafeq  18883  psrvscafval  19390  cnfldsub  19774  ipfeq  19995  mdetunilem7  20424  madurid  20450  cnmpt22f  21478  cnmptcom  21481  xkocnv  21617  qustgplem  21924  stdbdxmet  22320  iimulcn  22737  rrxds  23181  rrxmfval  23189  cnnvm  27537  ofpreima  29465  ressplusf  29650  matmpt2  29869  mndpluscn  29972  rmulccn  29974  raddcn  29975  txsconnlem  31222  cvmlift2lem6  31290  cvmlift2lem7  31291  cvmlift2lem12  31296  unccur  33392  matunitlindflem1  33405  rngchomrnghmresALTV  41996