Theorem funeq 5138

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 3147	. . 3 |- ( A = B -> B C_ A )
2		funss 5137	. . 3 |- ( B C_ A -> ( Fun A -> Fun B ) )
3	1, 2	syl 14	. 2 |- ( A = B -> ( Fun A -> Fun B ) )
4		eqimss 3146	. . 3 |- ( A = B -> A C_ B )
5		funss 5137	. . 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 1331   C_ wss 3066   Fun wfun 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-in 3072  df-ss 3079  df-br 3925  df-opab 3985  df-rel 4541  df-cnv 4542  df-co 4543  df-fun 5120

This theorem is referenced by:  funeqi  5139  funeqd  5140  fununi  5186  funcnvuni  5187  cnvresid  5192  fneq1  5206  elpmg  6551  fundmeng  6694