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Theorem funss 5616
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 4938 . . 3  |-  ( A 
C_  B  ->  ( Rel  B  ->  Rel  A ) )
2 coss1 5006 . . . . 5  |-  ( A 
C_  B  ->  ( A  o.  `' A
)  C_  ( B  o.  `' A ) )
3 cnvss 5023 . . . . . 6  |-  ( A 
C_  B  ->  `' A  C_  `' B )
4 coss2 5007 . . . . . 6  |-  ( `' A  C_  `' B  ->  ( B  o.  `' A )  C_  ( B  o.  `' B
) )
53, 4syl 17 . . . . 5  |-  ( A 
C_  B  ->  ( B  o.  `' A
)  C_  ( B  o.  `' B ) )
62, 5sstrd 3474 . . . 4  |-  ( A 
C_  B  ->  ( A  o.  `' A
)  C_  ( B  o.  `' B ) )
7 sstr2 3471 . . . 4  |-  ( ( A  o.  `' A
)  C_  ( B  o.  `' B )  ->  (
( B  o.  `' B )  C_  _I  ->  ( A  o.  `' A )  C_  _I  ) )
86, 7syl 17 . . 3  |-  ( A 
C_  B  ->  (
( B  o.  `' B )  C_  _I  ->  ( A  o.  `' A )  C_  _I  ) )
91, 8anim12d 565 . 2  |-  ( A 
C_  B  ->  (
( Rel  B  /\  ( B  o.  `' B )  C_  _I  )  ->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  ) ) )
10 df-fun 5600 . 2  |-  ( Fun 
B  <->  ( Rel  B  /\  ( B  o.  `' B )  C_  _I  ) )
11 df-fun 5600 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  ( A  o.  `' A )  C_  _I  ) )
129, 10, 113imtr4g 273 1  |-  ( A 
C_  B  ->  ( Fun  B  ->  Fun  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    C_ wss 3436    _I cid 4760   `'ccnv 4849    o. ccom 4854   Rel wrel 4855   Fun wfun 5592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-in 3443  df-ss 3450  df-br 4421  df-opab 4480  df-rel 4857  df-cnv 4858  df-co 4859  df-fun 5600
This theorem is referenced by:  funeq  5617  funopab4  5633  funres  5637  fun0  5655  funcnvcnv  5656  funin  5665  funres11  5666  foimacnv  5845  funsssuppss  6949  strssd  15147  strle1  15209  xpsc0  15454  xpsc1  15455  pjpm  19258  frrlem5c  30515
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