Theorem funeq 5128

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

Assertion

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

Proof of Theorem funeq

Step	Hyp	Ref	Expression
1		funss 5127	. . . 4 |- (B C_ A -> (Fun A -> Fun B))
2		funss 5127	. . . 4 |- (A C_ B -> (Fun B -> Fun A))
3	1, 2	anim12i 549	. . 3 |- ((B C_ A /\ A C_ B) -> ((Fun A -> Fun B) /\ (Fun B -> Fun A)))
4	3	ancoms 439	. 2 |- ((A C_ B /\ B C_ A) -> ((Fun A -> Fun B) /\ (Fun B -> Fun A)))
5		eqss 3288	. 2 |- (A = B <-> (A C_ B /\ B C_ A))
6		dfbi2 609	. 2 |- ((Fun A <-> Fun B) <-> ((Fun A -> Fun B) /\ (Fun B -> Fun A)))
7	4, 5, 6	3imtr4i 257	1 |- (A = B -> (Fun A <-> Fun B))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   C_ wss 3258  Fun wfun 4776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-br 4641  df-co 4727  df-cnv 4786  df-fun 4790

This theorem is referenced by:  funeqi  5129  funeqd  5130  fununi  5161  funcnvuni  5162  cnvresid  5167  fneq1  5174  elfuns  5830  elfunsg  5831  elpmg  6014  fundmeng  6045