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Theorem ressval3d 12522
Description: Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)
Hypotheses
Ref Expression
ressval3d.r  |-  R  =  ( Ss  A )
ressval3d.b  |-  B  =  ( Base `  S
)
ressval3d.e  |-  E  =  ( Base `  ndx )
ressval3d.s  |-  ( ph  ->  S  e.  V )
ressval3d.f  |-  ( ph  ->  Fun  S )
ressval3d.d  |-  ( ph  ->  E  e.  dom  S
)
ressval3d.u  |-  ( ph  ->  A  C_  B )
Assertion
Ref Expression
ressval3d  |-  ( ph  ->  R  =  ( S sSet  <. E ,  A >. ) )

Proof of Theorem ressval3d
StepHypRef Expression
1 ressval3d.r . . 3  |-  R  =  ( Ss  A )
2 ressval3d.s . . . 4  |-  ( ph  ->  S  e.  V )
3 ressval3d.b . . . . . 6  |-  B  =  ( Base `  S
)
4 basfn 12511 . . . . . . 7  |-  Base  Fn  _V
52elexd 2750 . . . . . . 7  |-  ( ph  ->  S  e.  _V )
6 funfvex 5530 . . . . . . . 8  |-  ( ( Fun  Base  /\  S  e. 
dom  Base )  ->  ( Base `  S )  e. 
_V )
76funfni 5314 . . . . . . 7  |-  ( (
Base  Fn  _V  /\  S  e.  _V )  ->  ( Base `  S )  e. 
_V )
84, 5, 7sylancr 414 . . . . . 6  |-  ( ph  ->  ( Base `  S
)  e.  _V )
93, 8eqeltrid 2264 . . . . 5  |-  ( ph  ->  B  e.  _V )
10 ressval3d.u . . . . 5  |-  ( ph  ->  A  C_  B )
119, 10ssexd 4142 . . . 4  |-  ( ph  ->  A  e.  _V )
12 ressvalsets 12515 . . . 4  |-  ( ( S  e.  V  /\  A  e.  _V )  ->  ( Ss  A )  =  ( S sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  S
) ) >. )
)
132, 11, 12syl2anc 411 . . 3  |-  ( ph  ->  ( Ss  A )  =  ( S sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  S
) ) >. )
)
141, 13eqtrid 2222 . 2  |-  ( ph  ->  R  =  ( S sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  S
) ) >. )
)
15 ressval3d.e . . . . 5  |-  E  =  ( Base `  ndx )
1615a1i 9 . . . 4  |-  ( ph  ->  E  =  ( Base `  ndx ) )
17 df-ss 3142 . . . . . 6  |-  ( A 
C_  B  <->  ( A  i^i  B )  =  A )
1810, 17sylib 122 . . . . 5  |-  ( ph  ->  ( A  i^i  B
)  =  A )
193ineq2i 3333 . . . . 5  |-  ( A  i^i  B )  =  ( A  i^i  ( Base `  S ) )
2018, 19eqtr3di 2225 . . . 4  |-  ( ph  ->  A  =  ( A  i^i  ( Base `  S
) ) )
2116, 20opeq12d 3786 . . 3  |-  ( ph  -> 
<. E ,  A >.  = 
<. ( Base `  ndx ) ,  ( A  i^i  ( Base `  S
) ) >. )
2221oveq2d 5887 . 2  |-  ( ph  ->  ( S sSet  <. E ,  A >. )  =  ( S sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  S
) ) >. )
)
2314, 22eqtr4d 2213 1  |-  ( ph  ->  R  =  ( S sSet  <. E ,  A >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   _Vcvv 2737    i^i cin 3128    C_ wss 3129   <.cop 3595   dom cdm 4625   Fun wfun 5208    Fn wfn 5209   ` cfv 5214  (class class class)co 5871   ndxcnx 12450   sSet csts 12451   Basecbs 12453   ↾s cress 12454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7898  ax-resscn 7899  ax-1re 7901  ax-addrcl 7904
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-iota 5176  df-fun 5216  df-fn 5217  df-fv 5222  df-ov 5874  df-oprab 5875  df-mpo 5876  df-inn 8915  df-ndx 12456  df-slot 12457  df-base 12459  df-sets 12460  df-iress 12461
This theorem is referenced by: (None)
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