Theorem funeqd 5340

Description: Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)

Hypothesis

Ref	Expression
funeqd.1	|- ( ph -> A = B )

Assertion

Ref	Expression
funeqd	|- ( ph -> ( Fun A <-> Fun B ) )

Proof of Theorem funeqd

Step	Hyp	Ref	Expression
1		funeqd.1	. 2 |- ( ph -> A = B )
2		funeq 5338	. 2 |- ( A = B -> ( Fun A <-> Fun B ) )
3	1, 2	syl 14	1 |- ( ph -> ( Fun A <-> Fun B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   Fun wfun 5312

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3203  df-ss 3210  df-br 4084  df-opab 4146  df-rel 4726  df-cnv 4727  df-co 4728  df-fun 5320

This theorem is referenced by:  funopg  5352  funsng  5367  funcnvuni  5390  f1eq1  5528  funopsn  5819  frecuzrdgtclt  10655  fundm2domnop0  11080  shftfn  11351  ennnfonelemfun  13004  ennnfonelemf1  13005  isstruct2im  13058  isstruct2r  13059  structfung  13065  setsfun  13083  setsfun0  13084  strslfv3  13094  istrl  16129  funmptd  16250