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Theorem fsets 14672
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 3627 . . . . . 6  |-  ( A 
\  { X }
)  C_  A
2 fssres 5757 . . . . . 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 5296 . . . . . . . 8  |-  ( ( F  |`  A )  |`  ( _V  \  { X } ) )  =  ( F  |`  ( A  i^i  ( _V  \  { X } ) ) )
5 invdif 3746 . . . . . . . . 9  |-  ( A  i^i  ( _V  \  { X } ) )  =  ( A  \  { X } )
65reseq2i 5280 . . . . . . . 8  |-  ( F  |`  ( A  i^i  ( _V  \  { X }
) ) )  =  ( F  |`  ( A  \  { X }
) )
74, 6eqtri 2486 . . . . . . 7  |-  ( ( F  |`  A )  |`  ( _V  \  { X } ) )  =  ( F  |`  ( A  \  { X }
) )
8 ffn 5737 . . . . . . . . 9  |-  ( F : A --> B  ->  F  Fn  A )
9 fnresdm 5696 . . . . . . . . 9  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
108, 9syl 16 . . . . . . . 8  |-  ( F : A --> B  -> 
( F  |`  A )  =  F )
1110reseq1d 5282 . . . . . . 7  |-  ( F : A --> B  -> 
( ( F  |`  A )  |`  ( _V  \  { X }
) )  =  ( F  |`  ( _V  \  { X } ) ) )
127, 11syl5reqr 2513 . . . . . 6  |-  ( F : A --> B  -> 
( F  |`  ( _V  \  { X }
) )  =  ( F  |`  ( A  \  { X } ) ) )
1312feq1d 5723 . . . . 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 6111 . . 3  |-  ( ( ( F  |`  ( _V  \  { X }
) ) : ( A  \  { X } ) --> B  /\  X  e.  A  /\  Y  e.  B )  ->  ( ( F  |`  ( _V  \  { X } ) )  u. 
{ <. X ,  Y >. } ) : A --> B )
1715, 16syl3an1 1261 . 2  |-  ( ( ( F  e.  V  /\  F : A --> B )  /\  X  e.  A  /\  Y  e.  B
)  ->  ( ( F  |`  ( _V  \  { X } ) )  u.  { <. X ,  Y >. } ) : A --> B )
18 simp1l 1020 . . 3  |-  ( ( ( F  e.  V  /\  F : A --> B )  /\  X  e.  A  /\  Y  e.  B
)  ->  F  e.  V )
19 simp3 998 . . 3  |-  ( ( ( F  e.  V  /\  F : A --> B )  /\  X  e.  A  /\  Y  e.  B
)  ->  Y  e.  B )
20 setsval 14669 . . . 4  |-  ( ( F  e.  V  /\  Y  e.  B )  ->  ( F sSet  <. X ,  Y >. )  =  ( ( F  |`  ( _V  \  { X }
) )  u.  { <. X ,  Y >. } ) )
2120feq1d 5723 . . 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 973    = wceq 1395    e. wcel 1819   _Vcvv 3109    \ cdif 3468    u. cun 3469    i^i cin 3470    C_ wss 3471   {csn 4032   <.cop 4038    |` cres 5010    Fn wfn 5589   -->wf 5590  (class class class)co 6296   sSet csts 14642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-sets 14650
This theorem is referenced by:  mdetunilem9  19249
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