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Theorem ressbas 14564
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 997 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  B  C_  A )
3 sseqin2 3702 . . . . . 6  |-  ( B 
C_  A  <->  ( A  i^i  B )  =  B )
42, 3sylib 196 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B )  =  B )
5 ressbas.r . . . . . . 7  |-  R  =  ( Ws  A )
65, 1ressid2 14562 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  R  =  W )
76fveq2d 5860 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( Base `  R )  =  ( Base `  W
) )
81, 4, 73eqtr4a 2510 . . . 4  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B )  =  ( Base `  R
) )
983expib 1200 . . 3  |-  ( B 
C_  A  ->  (
( W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B )  =  ( Base `  R ) ) )
10 simp2 998 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  W  e.  _V )
11 fvex 5866 . . . . . . . 8  |-  ( Base `  W )  e.  _V
121, 11eqeltri 2527 . . . . . . 7  |-  B  e. 
_V
1312inex2 4579 . . . . . 6  |-  ( A  i^i  B )  e. 
_V
14 baseid 14555 . . . . . . 7  |-  Base  = Slot  ( Base `  ndx )
1514setsid 14550 . . . . . 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 662 . . . . 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 14563 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  R  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
)
1817fveq2d 5860 . . . . 5  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( Base `  R
)  =  ( Base `  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
1916, 18eqtr4d 2487 . . . 4  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B )  =  ( Base `  R ) )
20193expib 1200 . . 3  |-  ( -.  B  C_  A  ->  ( ( W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B )  =  ( Base `  R ) ) )
219, 20pm2.61i 164 . 2  |-  ( ( W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B
)  =  ( Base `  R ) )
22 0fv 5889 . . . . 5  |-  ( (/) `  ( Base `  ndx ) )  =  (/)
23 0ex 4567 . . . . . 6  |-  (/)  e.  _V
2423, 14strfvn 14526 . . . . 5  |-  ( Base `  (/) )  =  (
(/) `  ( Base ` 
ndx ) )
25 in0 3797 . . . . 5  |-  ( A  i^i  (/) )  =  (/)
2622, 24, 253eqtr4ri 2483 . . . 4  |-  ( A  i^i  (/) )  =  (
Base `  (/) )
27 fvprc 5850 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
281, 27syl5eq 2496 . . . . 5  |-  ( -.  W  e.  _V  ->  B  =  (/) )
2928ineq2d 3685 . . . 4  |-  ( -.  W  e.  _V  ->  ( A  i^i  B )  =  ( A  i^i  (/) ) )
30 reldmress 14560 . . . . . . 7  |-  Rel  doms
3130ovprc1 6312 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  (/) )
325, 31syl5eq 2496 . . . . 5  |-  ( -.  W  e.  _V  ->  R  =  (/) )
3332fveq2d 5860 . . . 4  |-  ( -.  W  e.  _V  ->  (
Base `  R )  =  ( Base `  (/) ) )
3426, 29, 333eqtr4a 2510 . . 3  |-  ( -.  W  e.  _V  ->  ( A  i^i  B )  =  ( Base `  R
) )
3534adantr 465 . 2  |-  ( ( -.  W  e.  _V  /\  A  e.  V )  ->  ( A  i^i  B )  =  ( Base `  R ) )
3621, 35pm2.61ian 790 1  |-  ( A  e.  V  ->  ( A  i^i  B )  =  ( Base `  R
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   _Vcvv 3095    i^i cin 3460    C_ wss 3461   (/)c0 3770   <.cop 4020   ` cfv 5578  (class class class)co 6281   ndxcnx 14506   sSet csts 14507   Basecbs 14509   ↾s cress 14510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-i2m1 9563  ax-1ne0 9564  ax-rrecex 9567  ax-cnre 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-recs 7044  df-rdg 7078  df-nn 10543  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516
This theorem is referenced by:  ressbas2  14565  ressbasss  14566  ressress  14571  rescabs  15079  resscatc  15306  resscntz  16243  idrespermg  16310  opprsubg  17159  subrgpropd  17337  sralmod  17707  resstopn  19560  resstps  19561  ressuss  20639  ressxms  20901  ressms  20902  cphsubrglem  21497  resspos  27520  resstos  27521  xrge0base  27546  xrge00  27547  submomnd  27573  suborng  27678  lidlssbas  32435  lidlbas  32436  uzlidlring  32442  dmatALTbas  32732
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