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Theorem ressbas 13198
Description: Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.)
Hypotheses
Ref Expression
ressbas.r  |-  R  =  ( Ws  A )
ressbas.b  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
ressbas  |-  ( A  e.  V  ->  ( A  i^i  B )  =  ( Base `  R
) )

Proof of Theorem ressbas
StepHypRef Expression
1 ressbas.b . . . . 5  |-  B  =  ( Base `  W
)
2 simp1 955 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  B  C_  A )
3 sseqin2 3388 . . . . . 6  |-  ( B 
C_  A  <->  ( A  i^i  B )  =  B )
42, 3sylib 188 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B )  =  B )
5 ressbas.r . . . . . . 7  |-  R  =  ( Ws  A )
65, 1ressid2 13196 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  R  =  W )
76fveq2d 5529 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( Base `  R )  =  ( Base `  W
) )
81, 4, 73eqtr4a 2341 . . . 4  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B )  =  ( Base `  R
) )
983expib 1154 . . 3  |-  ( B 
C_  A  ->  (
( W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B )  =  ( Base `  R ) ) )
10 simp2 956 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  W  e.  _V )
11 fvex 5539 . . . . . . . 8  |-  ( Base `  W )  e.  _V
121, 11eqeltri 2353 . . . . . . 7  |-  B  e. 
_V
1312inex2 4156 . . . . . 6  |-  ( A  i^i  B )  e. 
_V
14 baseid 13190 . . . . . . 7  |-  Base  = Slot  ( Base `  ndx )
1514setsid 13187 . . . . . 6  |-  ( ( W  e.  _V  /\  ( A  i^i  B )  e.  _V )  -> 
( A  i^i  B
)  =  ( Base `  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
1610, 13, 15sylancl 643 . . . . 5  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B )  =  ( Base `  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
175, 1ressval2 13197 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  R  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
)
1817fveq2d 5529 . . . . 5  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( Base `  R
)  =  ( Base `  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
1916, 18eqtr4d 2318 . . . 4  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B )  =  ( Base `  R ) )
20193expib 1154 . . 3  |-  ( -.  B  C_  A  ->  ( ( W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B )  =  ( Base `  R ) ) )
219, 20pm2.61i 156 . 2  |-  ( ( W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B
)  =  ( Base `  R ) )
22 fv01 5559 . . . . 5  |-  ( (/) `  ( Base `  ndx ) )  =  (/)
23 0ex 4150 . . . . . 6  |-  (/)  e.  _V
2423, 14strfvn 13165 . . . . 5  |-  ( Base `  (/) )  =  (
(/) `  ( Base ` 
ndx ) )
25 in0 3480 . . . . 5  |-  ( A  i^i  (/) )  =  (/)
2622, 24, 253eqtr4ri 2314 . . . 4  |-  ( A  i^i  (/) )  =  (
Base `  (/) )
27 fvprc 5519 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
281, 27syl5eq 2327 . . . . 5  |-  ( -.  W  e.  _V  ->  B  =  (/) )
2928ineq2d 3370 . . . 4  |-  ( -.  W  e.  _V  ->  ( A  i^i  B )  =  ( A  i^i  (/) ) )
30 reldmress 13194 . . . . . . 7  |-  Rel  doms
3130ovprc1 5886 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  (/) )
325, 31syl5eq 2327 . . . . 5  |-  ( -.  W  e.  _V  ->  R  =  (/) )
3332fveq2d 5529 . . . 4  |-  ( -.  W  e.  _V  ->  (
Base `  R )  =  ( Base `  (/) ) )
3426, 29, 333eqtr4a 2341 . . 3  |-  ( -.  W  e.  _V  ->  ( A  i^i  B )  =  ( Base `  R
) )
3534adantr 451 . 2  |-  ( ( -.  W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B )  =  ( Base `  R ) )
3621, 35pm2.61ian 765 1  |-  ( A  e.  V  ->  ( A  i^i  B )  =  ( Base `  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   ` cfv 5255  (class class class)co 5858   ndxcnx 13145   sSet csts 13146   Basecbs 13148   ↾s cress 13149
This theorem is referenced by:  ressbas2  13199  ressbasss  13200  ressress  13205  rescabs  13710  resscatc  13937  resscntz  14807  opprsubg  15418  subrgpropd  15579  sralmod  15939  resstopn  16916  resstps  16917  ressxms  18071  ressms  18072  cphsubrglem  18613  xrge0base  23310  xrge00  23311
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
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-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-nn 9747  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155
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