Theorem funss 5126

Description: Subclass theorem for function predicate. (The proof was shortened by Mario Carneiro, 24-Jun-2014.) (Contributed by set.mm contributors, 16-Aug-1994.) (Revised by set.mm contributors, 24-Jun-2014.)

Assertion

Ref	Expression
funss	|- (A C_ B -> (Fun B -> Fun A))

Proof of Theorem funss

Step	Hyp	Ref	Expression
1		coss1 4872	. . . 4 |- (A C_ B -> (A o. `'A) C_ (B o. `'A))
2		cnvss 4885	. . . . 5 |- (A C_ B -> `'A C_ `'B)
3		coss2 4873	. . . . 5 |- (`'A C_ `'B -> (B o. `'A) C_ (B o. `'B))
4	2, 3	syl 15	. . . 4 |- (A C_ B -> (B o. `'A) C_ (B o. `'B))
5	1, 4	sstrd 3282	. . 3 |- (A C_ B -> (A o. `'A) C_ (B o. `'B))
6		sstr2 3279	. . 3 |- ((A o. `'A) C_ (B o. `'B) -> ((B o. `'B) C_ _I -> (A o. `'A) C_ _I ))
7	5, 6	syl 15	. 2 |- (A C_ B -> ((B o. `'B) C_ _I -> (A o. `'A) C_ _I ))
8		df-fun 4789	. 2 |- (Fun B <-> (B o. `'B) C_ _I )
9		df-fun 4789	. 2 |- (Fun A <-> (A o. `'A) C_ _I )
10	7, 8, 9	3imtr4g 261	1 |- (A C_ B -> (Fun B -> Fun A))

Syntax hints:   -> wi 4   C_ wss 3257   o. ccom 4721   _I cid 4763  `'ccnv 4771  Fun wfun 4775

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-opab 4623  df-br 4640  df-co 4726  df-cnv 4785  df-fun 4789

This theorem is referenced by:  funeq  5127  funopab4  5141  funres  5143  funin  5163  funres11  5164  foimacnv  5303