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Theorem fsets 14304
Description: The structure replacement function is a function. (Contributed by SO, 12-Jul-2018.)
Assertion
Ref Expression
fsets  |-  ( ( ( F  e.  V  /\  F : A --> B )  /\  X  e.  A  /\  Y  e.  B
)  ->  ( F sSet  <. X ,  Y >. ) : A --> B )

Proof of Theorem fsets
StepHypRef Expression
1 difss 3583 . . . . . 6  |-  ( A 
\  { X }
)  C_  A
2 fssres 5678 . . . . . 6  |-  ( ( F : A --> B  /\  ( A  \  { X } )  C_  A
)  ->  ( F  |`  ( A  \  { X } ) ) : ( A  \  { X } ) --> B )
31, 2mpan2 671 . . . . 5  |-  ( F : A --> B  -> 
( F  |`  ( A  \  { X }
) ) : ( A  \  { X } ) --> B )
4 resres 5223 . . . . . . . 8  |-  ( ( F  |`  A )  |`  ( _V  \  { X } ) )  =  ( F  |`  ( A  i^i  ( _V  \  { X } ) ) )
5 invdif 3691 . . . . . . . . 9  |-  ( A  i^i  ( _V  \  { X } ) )  =  ( A  \  { X } )
65reseq2i 5207 . . . . . . . 8  |-  ( F  |`  ( A  i^i  ( _V  \  { X }
) ) )  =  ( F  |`  ( A  \  { X }
) )
74, 6eqtri 2480 . . . . . . 7  |-  ( ( F  |`  A )  |`  ( _V  \  { X } ) )  =  ( F  |`  ( A  \  { X }
) )
8 ffn 5659 . . . . . . . . 9  |-  ( F : A --> B  ->  F  Fn  A )
9 fnresdm 5620 . . . . . . . . 9  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
108, 9syl 16 . . . . . . . 8  |-  ( F : A --> B  -> 
( F  |`  A )  =  F )
1110reseq1d 5209 . . . . . . 7  |-  ( F : A --> B  -> 
( ( F  |`  A )  |`  ( _V  \  { X }
) )  =  ( F  |`  ( _V  \  { X } ) ) )
127, 11syl5reqr 2507 . . . . . 6  |-  ( F : A --> B  -> 
( F  |`  ( _V  \  { X }
) )  =  ( F  |`  ( A  \  { X } ) ) )
1312feq1d 5646 . . . . 5  |-  ( F : A --> B  -> 
( ( F  |`  ( _V  \  { X } ) ) : ( A  \  { X } ) --> B  <->  ( F  |`  ( A  \  { X } ) ) : ( A  \  { X } ) --> B ) )
143, 13mpbird 232 . . . 4  |-  ( F : A --> B  -> 
( F  |`  ( _V  \  { X }
) ) : ( A  \  { X } ) --> B )
1514adantl 466 . . 3  |-  ( ( F  e.  V  /\  F : A --> B )  ->  ( F  |`  ( _V  \  { X } ) ) : ( A  \  { X } ) --> B )
16 fsnunf2 6018 . . 3  |-  ( ( ( F  |`  ( _V  \  { X }
) ) : ( A  \  { X } ) --> B  /\  X  e.  A  /\  Y  e.  B )  ->  ( ( F  |`  ( _V  \  { X } ) )  u. 
{ <. X ,  Y >. } ) : A --> B )
1715, 16syl3an1 1252 . 2  |-  ( ( ( F  e.  V  /\  F : A --> B )  /\  X  e.  A  /\  Y  e.  B
)  ->  ( ( F  |`  ( _V  \  { X } ) )  u.  { <. X ,  Y >. } ) : A --> B )
18 simp1l 1012 . . 3  |-  ( ( ( F  e.  V  /\  F : A --> B )  /\  X  e.  A  /\  Y  e.  B
)  ->  F  e.  V )
19 simp3 990 . . 3  |-  ( ( ( F  e.  V  /\  F : A --> B )  /\  X  e.  A  /\  Y  e.  B
)  ->  Y  e.  B )
20 setsval 14302 . . . 4  |-  ( ( F  e.  V  /\  Y  e.  B )  ->  ( F sSet  <. X ,  Y >. )  =  ( ( F  |`  ( _V  \  { X }
) )  u.  { <. X ,  Y >. } ) )
2120feq1d 5646 . . 3  |-  ( ( F  e.  V  /\  Y  e.  B )  ->  ( ( F sSet  <. X ,  Y >. ) : A --> B  <->  ( ( F  |`  ( _V  \  { X } ) )  u.  { <. X ,  Y >. } ) : A --> B ) )
2218, 19, 21syl2anc 661 . 2  |-  ( ( ( F  e.  V  /\  F : A --> B )  /\  X  e.  A  /\  Y  e.  B
)  ->  ( ( F sSet  <. X ,  Y >. ) : A --> B  <->  ( ( F  |`  ( _V  \  { X } ) )  u.  { <. X ,  Y >. } ) : A --> B ) )
2317, 22mpbird 232 1  |-  ( ( ( F  e.  V  /\  F : A --> B )  /\  X  e.  A  /\  Y  e.  B
)  ->  ( F sSet  <. X ,  Y >. ) : A --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3070    \ cdif 3425    u. cun 3426    i^i cin 3427    C_ wss 3428   {csn 3977   <.cop 3983    |` cres 4942    Fn wfn 5513   -->wf 5514  (class class class)co 6192   sSet csts 14276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-sets 14284
This theorem is referenced by:  mdetunilem9  18544
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