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Theorem funss 5112
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 4596 . . 3  |-  ( A 
C_  B  ->  ( Rel  B  ->  Rel  A ) )
2 coss1 4664 . . . . 5  |-  ( A 
C_  B  ->  ( A  o.  `' A
)  C_  ( B  o.  `' A ) )
3 cnvss 4682 . . . . . 6  |-  ( A 
C_  B  ->  `' A  C_  `' B )
4 coss2 4665 . . . . . 6  |-  ( `' A  C_  `' B  ->  ( B  o.  `' A )  C_  ( B  o.  `' B
) )
53, 4syl 14 . . . . 5  |-  ( A 
C_  B  ->  ( B  o.  `' A
)  C_  ( B  o.  `' B ) )
62, 5sstrd 3077 . . . 4  |-  ( A 
C_  B  ->  ( A  o.  `' A
)  C_  ( B  o.  `' B ) )
7 sstr2 3074 . . . 4  |-  ( ( A  o.  `' A
)  C_  ( B  o.  `' B )  ->  (
( B  o.  `' B )  C_  _I  ->  ( A  o.  `' A )  C_  _I  ) )
86, 7syl 14 . . 3  |-  ( A 
C_  B  ->  (
( B  o.  `' B )  C_  _I  ->  ( A  o.  `' A )  C_  _I  ) )
91, 8anim12d 333 . 2  |-  ( A 
C_  B  ->  (
( Rel  B  /\  ( B  o.  `' B )  C_  _I  )  ->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  ) ) )
10 df-fun 5095 . 2  |-  ( Fun 
B  <->  ( Rel  B  /\  ( B  o.  `' B )  C_  _I  ) )
11 df-fun 5095 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  ) )
129, 10, 113imtr4g 204 1  |-  ( A 
C_  B  ->  ( Fun  B  ->  Fun  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    C_ wss 3041    _I cid 4180   `'ccnv 4508    o. ccom 4513   Rel wrel 4514   Fun wfun 5087
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-in 3047  df-ss 3054  df-br 3900  df-opab 3960  df-rel 4516  df-cnv 4517  df-co 4518  df-fun 5095
This theorem is referenced by:  funeq  5113  funopab4  5130  funres  5134  fun0  5151  funcnvcnv  5152  funin  5164  funres11  5165  foimacnv  5353  tfrlemibfn  6193  tfr1onlembfn  6209  tfrcllembfn  6222  strslssd  11932  strle1g  11976
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