Theorem fneq1 5363

Description: Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
fneq1	|- ( F = G -> ( F Fn A <-> G Fn A ) )

Proof of Theorem fneq1

Step	Hyp	Ref	Expression
1		funeq 5292	. . 3 |- ( F = G -> ( Fun F <-> Fun G ) )
2		dmeq 4879	. . . 4 |- ( F = G -> dom F = dom G )
3	2	eqeq1d 2214	. . 3 |- ( F = G -> ( dom F = A <-> dom G = A ) )
4	1, 3	anbi12d 473	. 2 |- ( F = G -> ( ( Fun F /\ dom F = A ) <-> ( Fun G /\ dom G = A ) ) )
5		df-fn 5275	. 2 |- ( F Fn A <-> ( Fun F /\ dom F = A ) )
6		df-fn 5275	. 2 |- ( G Fn A <-> ( Fun G /\ dom G = A ) )
7	4, 5, 6	3bitr4g 223	1 |- ( F = G -> ( F Fn A <-> G Fn A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   dom cdm 4676   Fun wfun 5266   Fn wfn 5267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-fun 5274  df-fn 5275

This theorem is referenced by:  fneq1d  5365  fneq1i  5369  fn0  5397  feq1  5410  foeq1  5496  f1ocnv  5537  mpteqb  5672  eufnfv  5817  uchoice  6225  tfr0dm  6410  tfrlemiex  6419  tfr1onlemsucfn  6428  tfr1onlemsucaccv  6429  tfr1onlembxssdm  6431  tfr1onlembfn  6432  tfr1onlemex  6435  tfr1onlemaccex  6436  tfr1onlemres  6437  mapval2  6767  elixp2  6791  ixpfn  6793  elixpsn  6824  cc2lem  7380  cc3  7382  lmodfopnelem1  14119