Theorem funeq 5377

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 3297	. . 3 |- ( A = B -> B C_ A )
2		funss 5376	. . 3 |- ( B C_ A -> ( Fun A -> Fun B ) )
3	1, 2	syl 14	. 2 |- ( A = B -> ( Fun A -> Fun B ) )
4		eqimss 3296	. . 3 |- ( A = B -> A C_ B )
5		funss 5376	. . 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 1398   C_ wss 3214   Fun wfun 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-in 3220  df-ss 3227  df-br 4115  df-opab 4177  df-rel 4761  df-cnv 4762  df-co 4763  df-fun 5359

This theorem is referenced by:  funeqi  5378  funeqd  5379  fununi  5429  funcnvuni  5430  cnvresid  5435  fneq1  5449  funop  5866  elpmg  6911  fundmeng  7061  isfsupp  7255  usgredgop  16294