Theorem funeqd 5348

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 5346	. 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 1397   Fun wfun 5320

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-in 3206  df-ss 3213  df-br 4089  df-opab 4151  df-rel 4732  df-cnv 4733  df-co 4734  df-fun 5328

This theorem is referenced by:  funopg  5360  funsng  5376  funcnvuni  5399  f1eq1  5537  f1ssf1  5615  funopsn  5830  frecuzrdgtclt  10684  fundm2domnop0  11113  shftfn  11402  ennnfonelemfun  13056  ennnfonelemf1  13057  isstruct2im  13110  isstruct2r  13111  structfung  13117  setsfun  13135  setsfun0  13136  strslfv3  13146  uhgrspansubgrlem  16146  p1evtxdeqfilem  16181  istrl  16255  trlsegvdeglem2  16331  trlsegvdeglem3  16332  funmptd  16450