Theorem ressbasd 13398

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 4264	. . . 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 13388	. . . 4 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
6	5	setsslid 13381	. . 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 415	. 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 3432	. 2 |- ( ph -> ( A i^i B ) = ( A i^i ( Base ` W ) ) )
10		ressbasd.r	. . . 4 |- ( ph -> R = ( W ↾s A ) )
11		ressvalsets 13395	. . . . 5 |- ( ( W e. X /\ A e. V ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
12	1, 2, 11	syl2anc 415	. . . 4 |- ( ph -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
13	10, 12	eqtrd 2271	. . 3 |- ( ph -> R = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
14	13	fveq2d 5694	. 2 |- ( ph -> ( Base ` R ) = ( Base ` ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
15	7, 9, 14	3eqtr4d 2281	1 |- ( ph -> ( A i^i B ) = ( Base ` R ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   <.cop 3708   ` cfv 5372  (class class class)co 6075   ndxcnx 13327   sSet csts 13328   Basecbs 13330   ↾_s cress 13331

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338

This theorem is referenced by:  ressbas2d  13399  ressbasssd  13400  ressbasid  13401  ressressg  13406  grpressid  13843  opprsubgg  14363  subrngpropd  14497  subrgpropd  14534  sralmod  14759  lidlbas  14787