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Theorem ressbas 13245
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 3422 . . . . . 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 13243 . . . . . 6  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  R  =  W )
76fveq2d 5567 . . . . 5  |-  ( ( B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( Base `  R )  =  ( Base `  W
) )
81, 4, 73eqtr4a 2374 . . . 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 5577 . . . . . . . 8  |-  ( Base `  W )  e.  _V
121, 11eqeltri 2386 . . . . . . 7  |-  B  e. 
_V
1312inex2 4193 . . . . . 6  |-  ( A  i^i  B )  e. 
_V
14 baseid 13237 . . . . . . 7  |-  Base  = Slot  ( Base `  ndx )
1514setsid 13234 . . . . . 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 13244 . . . . . 6  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  R  =  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B ) >. )
)
1817fveq2d 5567 . . . . 5  |-  ( ( -.  B  C_  A  /\  W  e.  _V  /\  A  e.  V )  ->  ( Base `  R
)  =  ( Base `  ( W sSet  <. ( Base `  ndx ) ,  ( A  i^i  B
) >. ) ) )
1916, 18eqtr4d 2351 . . . 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 5597 . . . . 5  |-  ( (/) `  ( Base `  ndx ) )  =  (/)
23 0ex 4187 . . . . . 6  |-  (/)  e.  _V
2423, 14strfvn 13212 . . . . 5  |-  ( Base `  (/) )  =  (
(/) `  ( Base ` 
ndx ) )
25 in0 3514 . . . . 5  |-  ( A  i^i  (/) )  =  (/)
2622, 24, 253eqtr4ri 2347 . . . 4  |-  ( A  i^i  (/) )  =  (
Base `  (/) )
27 fvprc 5557 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
281, 27syl5eq 2360 . . . . 5  |-  ( -.  W  e.  _V  ->  B  =  (/) )
2928ineq2d 3404 . . . 4  |-  ( -.  W  e.  _V  ->  ( A  i^i  B )  =  ( A  i^i  (/) ) )
30 reldmress 13241 . . . . . . 7  |-  Rel  doms
3130ovprc1 5928 . . . . . 6  |-  ( -.  W  e.  _V  ->  ( Ws  A )  =  (/) )
325, 31syl5eq 2360 . . . . 5  |-  ( -.  W  e.  _V  ->  R  =  (/) )
3332fveq2d 5567 . . . 4  |-  ( -.  W  e.  _V  ->  (
Base `  R )  =  ( Base `  (/) ) )
3426, 29, 333eqtr4a 2374 . . 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 1633    e. wcel 1701   _Vcvv 2822    i^i cin 3185    C_ wss 3186   (/)c0 3489   <.cop 3677   ` cfv 5292  (class class class)co 5900   ndxcnx 13192   sSet csts 13193   Basecbs 13195   ↾s cress 13196
This theorem is referenced by:  ressbas2  13246  ressbasss  13247  ressress  13252  rescabs  13759  resscatc  13986  resscntz  14856  opprsubg  15467  subrgpropd  15628  sralmod  15988  resstopn  16972  resstps  16973  ressxms  18123  ressms  18124  cphsubrglem  18666  xrge0base  23345  xrge00  23346  ressuss  23460
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-i2m1 8850  ax-1ne0 8851  ax-rrecex 8854  ax-cnre 8855
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-recs 6430  df-rdg 6465  df-nn 9792  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202
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