Theorem ressbasd 13014

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 4196	. . . 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 13004	. . . 4 |- ( Base = Slot ( Base ` ndx ) /\ ( Base ` ndx ) e. NN )
6	5	setsslid 12998	. . 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 3382	. 2 |- ( ph -> ( A i^i B ) = ( A i^i ( Base ` W ) ) )
10		ressbasd.r	. . . 4 |- ( ph -> R = ( W ↾s A ) )
11		ressvalsets 13011	. . . . 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 2240	. . 3 |- ( ph -> R = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
14	13	fveq2d 5603	. 2 |- ( ph -> ( Base ` R ) = ( Base ` ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
15	7, 9, 14	3eqtr4d 2250	1 |- ( ph -> ( A i^i B ) = ( Base ` R ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   _Vcvv 2776   i^i cin 3173   <.cop 3646   ` cfv 5290  (class class class)co 5967   ndxcnx 12944   sSet csts 12945   Basecbs 12947   ↾_s cress 12948

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-inn 9072  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955

This theorem is referenced by:  ressbas2d  13015  ressbasssd  13016  ressbasid  13017  ressressg  13022  grpressid  13508  opprsubgg  13961  subrngpropd  14093  subrgpropd  14130  sralmod  14327  lidlbas  14355