Theorem funeq 5099

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 3116	. . 3 |- ( A = B -> B C_ A )
2		funss 5098	. . 3 |- ( B C_ A -> ( Fun A -> Fun B ) )
3	1, 2	syl 14	. 2 |- ( A = B -> ( Fun A -> Fun B ) )
4		eqimss 3115	. . 3 |- ( A = B -> A C_ B )
5		funss 5098	. . 3 |- ( A C_ B -> ( Fun B -> Fun A ) )
6	4, 5	syl 14	. 2 |- ( A = B -> ( Fun B -> Fun A ) )
7	3, 6	impbid 128	1 |- ( A = B -> ( Fun A <-> Fun B ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1312   C_ wss 3035   Fun wfun 5073

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-in 3041  df-ss 3048  df-br 3894  df-opab 3948  df-rel 4504  df-cnv 4505  df-co 4506  df-fun 5081

This theorem is referenced by:  funeqi  5100  funeqd  5101  fununi  5147  funcnvuni  5148  cnvresid  5153  fneq1  5167  elpmg  6510  fundmeng  6653