Theorem ressbasd 12688

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 4166	. . . 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 12678	. . . 4 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
6	5	setsslid 12672	. . 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 3361	. 2 |- ( ph -> ( A i^i B ) = ( A i^i ( Base ` W ) ) )
10		ressbasd.r	. . . 4 |- ( ph -> R = ( W ↾s A ) )
11		ressvalsets 12685	. . . . 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 2226	. . 3 |- ( ph -> R = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
14	13	fveq2d 5559	. 2 |- ( ph -> ( Base ` R ) = ( Base ` ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
15	7, 9, 14	3eqtr4d 2236	1 |- ( ph -> ( A i^i B ) = ( Base ` R ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   _Vcvv 2760   i^i cin 3153   <.cop 3622   ` cfv 5255  (class class class)co 5919   ndxcnx 12618   sSet csts 12619   Basecbs 12621   ↾_s cress 12622

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-inn 8985  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629

This theorem is referenced by:  ressbas2d  12689  ressbasssd  12690  ressbasid  12691  ressressg  12696  grpressid  13136  opprsubgg  13583  subrngpropd  13715  subrgpropd  13752  sralmod  13949  lidlbas  13977