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Theorem funss 5289
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 4791 . . 3  |-  ( A 
C_  B  ->  ( Rel  B  ->  Rel  A ) )
2 coss1 4855 . . . . 5  |-  ( A 
C_  B  ->  ( A  o.  `' A
)  C_  ( B  o.  `' A ) )
3 cnvss 4870 . . . . . 6  |-  ( A 
C_  B  ->  `' A  C_  `' B )
4 coss2 4856 . . . . . 6  |-  ( `' A  C_  `' B  ->  ( B  o.  `' A )  C_  ( B  o.  `' B
) )
53, 4syl 15 . . . . 5  |-  ( A 
C_  B  ->  ( B  o.  `' A
)  C_  ( B  o.  `' B ) )
62, 5sstrd 3202 . . . 4  |-  ( A 
C_  B  ->  ( A  o.  `' A
)  C_  ( B  o.  `' B ) )
7 sstr2 3199 . . . 4  |-  ( ( A  o.  `' A
)  C_  ( B  o.  `' B )  ->  (
( B  o.  `' B )  C_  _I  ->  ( A  o.  `' A )  C_  _I  ) )
86, 7syl 15 . . 3  |-  ( A 
C_  B  ->  (
( B  o.  `' B )  C_  _I  ->  ( A  o.  `' A )  C_  _I  ) )
91, 8anim12d 546 . 2  |-  ( A 
C_  B  ->  (
( Rel  B  /\  ( B  o.  `' B )  C_  _I  )  ->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  ) ) )
10 df-fun 5273 . 2  |-  ( Fun 
B  <->  ( Rel  B  /\  ( B  o.  `' B )  C_  _I  ) )
11 df-fun 5273 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  ) )
129, 10, 113imtr4g 261 1  |-  ( A 
C_  B  ->  ( Fun  B  ->  Fun  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    C_ wss 3165    _I cid 4320   `'ccnv 4704    o. ccom 4709   Rel wrel 4710   Fun wfun 5265
This theorem is referenced by:  funeq  5290  funopab4  5305  funres  5309  fun0  5323  funcnvcnv  5324  funin  5335  funres11  5336  foimacnv  5506  strssd  13198  strle1  13255  xpsc0  13478  xpsc1  13479  pjpm  16624  frrlem5c  24358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-in 3172  df-ss 3179  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-fun 5273
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