Theorem funeqd 5294

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 5292	. 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 1373   Fun wfun 5266

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-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-in 3172  df-ss 3179  df-br 4046  df-opab 4107  df-rel 4683  df-cnv 4684  df-co 4685  df-fun 5274

This theorem is referenced by:  funopg  5306  funsng  5321  funcnvuni  5344  f1eq1  5478  funopsn  5764  frecuzrdgtclt  10568  fundm2domnop0  10992  shftfn  11168  ennnfonelemfun  12821  ennnfonelemf1  12822  isstruct2im  12875  isstruct2r  12876  structfung  12882  setsfun  12900  setsfun0  12901  strslfv3  12911  funmptd  15776