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Theorem funss 5597
Description: Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
Assertion
Ref Expression
funss  |-  ( A 
C_  B  ->  ( Fun  B  ->  Fun  A ) )

Proof of Theorem funss
StepHypRef Expression
1 relss 5081 . . 3  |-  ( A 
C_  B  ->  ( Rel  B  ->  Rel  A ) )
2 coss1 5149 . . . . 5  |-  ( A 
C_  B  ->  ( A  o.  `' A
)  C_  ( B  o.  `' A ) )
3 cnvss 5166 . . . . . 6  |-  ( A 
C_  B  ->  `' A  C_  `' B )
4 coss2 5150 . . . . . 6  |-  ( `' A  C_  `' B  ->  ( B  o.  `' A )  C_  ( B  o.  `' B
) )
53, 4syl 16 . . . . 5  |-  ( A 
C_  B  ->  ( B  o.  `' A
)  C_  ( B  o.  `' B ) )
62, 5sstrd 3507 . . . 4  |-  ( A 
C_  B  ->  ( A  o.  `' A
)  C_  ( B  o.  `' B ) )
7 sstr2 3504 . . . 4  |-  ( ( A  o.  `' A
)  C_  ( B  o.  `' B )  ->  (
( B  o.  `' B )  C_  _I  ->  ( A  o.  `' A )  C_  _I  ) )
86, 7syl 16 . . 3  |-  ( A 
C_  B  ->  (
( B  o.  `' B )  C_  _I  ->  ( A  o.  `' A )  C_  _I  ) )
91, 8anim12d 563 . 2  |-  ( A 
C_  B  ->  (
( Rel  B  /\  ( B  o.  `' B )  C_  _I  )  ->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  ) ) )
10 df-fun 5581 . 2  |-  ( Fun 
B  <->  ( Rel  B  /\  ( B  o.  `' B )  C_  _I  ) )
11 df-fun 5581 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  ) )
129, 10, 113imtr4g 270 1  |-  ( A 
C_  B  ->  ( Fun  B  ->  Fun  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    C_ wss 3469    _I cid 4783   `'ccnv 4991    o. ccom 4996   Rel wrel 4997   Fun wfun 5573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-in 3476  df-ss 3483  df-br 4441  df-opab 4499  df-rel 4999  df-cnv 5000  df-co 5001  df-fun 5581
This theorem is referenced by:  funeq  5598  funopab4  5614  funres  5618  fun0  5636  funcnvcnv  5637  funin  5646  funres11  5647  foimacnv  5824  funsssuppss  6916  strssd  14515  strle1  14575  xpsc0  14804  xpsc1  14805  pjpm  18499  frrlem5c  28956
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