Theorem ressbasd 12496

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 4136	. . . 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 12488	. . . 4 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
6	5	setsslid 12482	. . 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 3336	. 2 |- ( ph -> ( A i^i B ) = ( A i^i ( Base ` W ) ) )
10		ressbasd.r	. . . 4 |- ( ph -> R = ( W ↾s A ) )
11		ressvalsets 12493	. . . . 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 2210	. . 3 |- ( ph -> R = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
14	13	fveq2d 5514	. 2 |- ( ph -> ( Base ` R ) = ( Base ` ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
15	7, 9, 14	3eqtr4d 2220	1 |- ( ph -> ( A i^i B ) = ( Base ` R ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2737   i^i cin 3128   <.cop 3594   ` cfv 5211  (class class class)co 5868   ndxcnx 12429   sSet csts 12430   Basecbs 12432   ↾_s cress 12433

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-inn 8896  df-ndx 12435  df-slot 12436  df-base 12438  df-sets 12439  df-iress 12440

This theorem is referenced by:  ressbas2d  12497  ressbasssd  12498  ressressg  12503