Theorem funeq 5310

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 3256	. . 3 |- ( A = B -> B C_ A )
2		funss 5309	. . 3 |- ( B C_ A -> ( Fun A -> Fun B ) )
3	1, 2	syl 14	. 2 |- ( A = B -> ( Fun A -> Fun B ) )
4		eqimss 3255	. . 3 |- ( A = B -> A C_ B )
5		funss 5309	. . 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 1373   C_ wss 3174   Fun wfun 5284

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-in 3180  df-ss 3187  df-br 4060  df-opab 4122  df-rel 4700  df-cnv 4701  df-co 4702  df-fun 5292

This theorem is referenced by:  funeqi  5311  funeqd  5312  fununi  5361  funcnvuni  5362  cnvresid  5367  fneq1  5381  funop  5786  elpmg  6774  fundmeng  6923