Theorem funeqd 5250

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 5248	. 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 1363   Fun wfun 5222

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-in 3147  df-ss 3154  df-br 4016  df-opab 4077  df-rel 4645  df-cnv 4646  df-co 4647  df-fun 5230

This theorem is referenced by:  funopg  5262  funsng  5274  funcnvuni  5297  f1eq1  5428  frecuzrdgtclt  10435  shftfn  10847  ennnfonelemfun  12432  ennnfonelemf1  12433  isstruct2im  12486  isstruct2r  12487  structfung  12493  setsfun  12511  setsfun0  12512  funmptd  14851