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Theorem resslem 13201
Description: Other elements of a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
resslem.r  |-  R  =  ( Ws  A )
resslem.e  |-  C  =  ( E `  W
)
resslem.f  |-  E  = Slot 
N
resslem.n  |-  N  e.  NN
resslem.b  |-  1  <  N
Assertion
Ref Expression
resslem  |-  ( A  e.  V  ->  C  =  ( E `  R ) )

Proof of Theorem resslem
StepHypRef Expression
1 resslem.r . . . . . . 7  |-  R  =  ( Ws  A )
2 eqid 2283 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
31, 2ressid2 13196 . . . . . 6  |-  ( ( ( Base `  W
)  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  R  =  W )
43fveq2d 5529 . . . . 5  |-  ( ( ( Base `  W
)  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( E `  R )  =  ( E `  W ) )
543expib 1154 . . . 4  |-  ( (
Base `  W )  C_  A  ->  ( ( W  e.  _V  /\  A  e.  V )  ->  ( E `  R )  =  ( E `  W ) ) )
61, 2ressval2 13197 . . . . . . 7  |-  ( ( -.  ( Base `  W
)  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  R  =  ( W sSet  <. (
Base `  ndx ) ,  ( A  i^i  ( Base `  W ) )
>. ) )
76fveq2d 5529 . . . . . 6  |-  ( ( -.  ( Base `  W
)  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( E `  R )  =  ( E `  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W ) ) >.
) ) )
8 resslem.f . . . . . . . 8  |-  E  = Slot 
N
9 resslem.n . . . . . . . 8  |-  N  e.  NN
108, 9ndxid 13169 . . . . . . 7  |-  E  = Slot  ( E `  ndx )
118, 9ndxarg 13168 . . . . . . . . 9  |-  ( E `
 ndx )  =  N
12 1re 8837 . . . . . . . . . 10  |-  1  e.  RR
13 resslem.b . . . . . . . . . 10  |-  1  <  N
1412, 13gtneii 8930 . . . . . . . . 9  |-  N  =/=  1
1511, 14eqnetri 2463 . . . . . . . 8  |-  ( E `
 ndx )  =/=  1
16 basendx 13193 . . . . . . . 8  |-  ( Base `  ndx )  =  1
1715, 16neeqtrri 2469 . . . . . . 7  |-  ( E `
 ndx )  =/=  ( Base `  ndx )
1810, 17setsnid 13188 . . . . . 6  |-  ( E `
 W )  =  ( E `  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  ( Base `  W
) ) >. )
)
197, 18syl6eqr 2333 . . . . 5  |-  ( ( -.  ( Base `  W
)  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( E `  R )  =  ( E `  W ) )
20193expib 1154 . . . 4  |-  ( -.  ( Base `  W
)  C_  A  ->  ( ( W  e.  _V  /\  A  e.  V )  ->  ( E `  R )  =  ( E `  W ) ) )
215, 20pm2.61i 156 . . 3  |-  ( ( W  e.  _V  /\  A  e.  V )  ->  ( E `  R
)  =  ( E `
 W ) )
22 reldmress 13194 . . . . . . . . 9  |-  Rel  doms
2322ovprc1 5886 . . . . . . . 8  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  (/) )
241, 23syl5eq 2327 . . . . . . 7  |-  ( -.  W  e.  _V  ->  R  =  (/) )
2524fveq2d 5529 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( E `  R )  =  ( E `  (/) ) )
268str0 13184 . . . . . 6  |-  (/)  =  ( E `  (/) )
2725, 26syl6eqr 2333 . . . . 5  |-  ( -.  W  e.  _V  ->  ( E `  R )  =  (/) )
28 fvprc 5519 . . . . 5  |-  ( -.  W  e.  _V  ->  ( E `  W )  =  (/) )
2927, 28eqtr4d 2318 . . . 4  |-  ( -.  W  e.  _V  ->  ( E `  R )  =  ( E `  W ) )
3029adantr 451 . . 3  |-  ( ( -.  W  e.  _V  /\  A  e.  V )  ->  ( E `  R )  =  ( E `  W ) )
3121, 30pm2.61ian 765 . 2  |-  ( A  e.  V  ->  ( E `  R )  =  ( E `  W ) )
32 resslem.e . 2  |-  C  =  ( E `  W
)
3331, 32syl6reqr 2334 1  |-  ( A  e.  V  ->  C  =  ( E `  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   <.cop 3643   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   1c1 8738    < clt 8867   NNcn 9746   ndxcnx 13145   sSet csts 13146  Slot cslot 13147   Basecbs 13148   ↾s cress 13149
This theorem is referenced by:  ressplusg  13250  ressmulr  13261  ressstarv  13262  resssca  13283  ressvsca  13284  resstset  13299  ressle  13306  ressds  13318  resshom  13323  ressco  13324  rescco  13709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-nn 9747  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155
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