Theorem ressbasd 13280

Description: Base set of a structure restriction. (Contributed by Stefan O'Rear, 26-Nov-2014.) (Proof shortened by AV, 7-Nov-2024.)

Hypotheses

Ref	Expression
ressbasd.r	|- ( ph -> R = ( W ↾s A ) )
ressbasd.b	|- ( ph -> B = ( Base ` W ) )
ressbasd.w	|- ( ph -> W e. X )
ressbasd.a	|- ( ph -> A e. V )

Assertion

Ref	Expression
ressbasd	|- ( ph -> ( A i^i B ) = ( Base ` R ) )

Proof of Theorem ressbasd

Step	Hyp	Ref	Expression
1		ressbasd.w	. . 3 |- ( ph -> W e. X )
2		ressbasd.a	. . . 4 |- ( ph -> A e. V )
3		inex1g 4246	. . . 4 |- ( A e. V -> ( A i^i ( Base ` W ) ) e. _V )
4	2, 3	syl 14	. . 3 |- ( ph -> ( A i^i ( Base ` W ) ) e. _V )
5		baseslid 13270	. . . 4 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
6	5	setsslid 13263	. . 3 |- ( ( W e. X /\ ( A i^i ( Base ` W ) ) e. _V ) -> ( A i^i ( Base ` W ) ) = ( Base ` ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
7	1, 4, 6	syl2anc 411	. 2 |- ( ph -> ( A i^i ( Base ` W ) ) = ( Base ` ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
8		ressbasd.b	. . 3 |- ( ph -> B = ( Base ` W ) )
9	8	ineq2d 3422	. 2 |- ( ph -> ( A i^i B ) = ( A i^i ( Base ` W ) ) )
10		ressbasd.r	. . . 4 |- ( ph -> R = ( W ↾s A ) )
11		ressvalsets 13277	. . . . 5 |- ( ( W e. X /\ A e. V ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
12	1, 2, 11	syl2anc 411	. . . 4 |- ( ph -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
13	10, 12	eqtrd 2265	. . 3 |- ( ph -> R = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
14	13	fveq2d 5674	. 2 |- ( ph -> ( Base ` R ) = ( Base ` ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
15	7, 9, 14	3eqtr4d 2275	1 |- ( ph -> ( A i^i B ) = ( Base ` R ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   _Vcvv 2813   i^i cin 3210   <.cop 3692   ` cfv 5352  (class class class)co 6050   ndxcnx 13209   sSet csts 13210   Basecbs 13212   ↾_s cress 13213

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220

This theorem is referenced by:  ressbas2d  13281  ressbasssd  13282  ressbasid  13283  ressressg  13288  grpressid  13774  opprsubgg  14228  subrngpropd  14361  subrgpropd  14398  sralmod  14598  lidlbas  14626