Theorem funeq 5338

Description: Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)

Assertion

Ref	Expression
funeq	|- ( A = B -> ( Fun A <-> Fun B ) )

Proof of Theorem funeq

Step	Hyp	Ref	Expression
1		eqimss2 3279	. . 3 |- ( A = B -> B C_ A )
2		funss 5337	. . 3 |- ( B C_ A -> ( Fun A -> Fun B ) )
3	1, 2	syl 14	. 2 |- ( A = B -> ( Fun A -> Fun B ) )
4		eqimss 3278	. . 3 |- ( A = B -> A C_ B )
5		funss 5337	. . 3 |- ( A C_ B -> ( Fun B -> Fun A ) )
6	4, 5	syl 14	. 2 |- ( A = B -> ( Fun B -> Fun A ) )
7	3, 6	impbid 129	1 |- ( A = B -> ( Fun A <-> Fun B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   C_ wss 3197   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:  funeqi  5339  funeqd  5340  fununi  5389  funcnvuni  5390  cnvresid  5395  fneq1  5409  funop  5818  elpmg  6811  fundmeng  6960  usgredgop  15971