Theorem resseqnbasd 13219

Description: The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)

Hypotheses

Ref	Expression
resseqnbas.r	|- R = ( W ↾s A )
resseqnbas.e	|- C = ( E ` W )
resseqnbasd.f	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
resseqnbas.n	|- ( E ` ndx ) =/= ( Base ` ndx )
resseqnbasd.w	|- ( ph -> W e. X )
resseqnbasd.a	|- ( ph -> A e. V )

Assertion

Ref	Expression
resseqnbasd	|- ( ph -> C = ( E ` R ) )

Proof of Theorem resseqnbasd

Step	Hyp	Ref	Expression
1		resseqnbas.e	. 2 |- C = ( E ` W )
2		resseqnbas.r	. . . . 5 |- R = ( W ↾s A )
3		resseqnbasd.w	. . . . . 6 |- ( ph -> W e. X )
4		resseqnbasd.a	. . . . . 6 |- ( ph -> A e. V )
5		ressvalsets 13210	. . . . . 6 |- ( ( W e. X /\ A e. V ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
6	3, 4, 5	syl2anc 411	. . . . 5 |- ( ph -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
7	2, 6	eqtrid 2276	. . . 4 |- ( ph -> R = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
8	7	fveq2d 5652	. . 3 |- ( ph -> ( E ` R ) = ( E ` ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
9		inex1g 4230	. . . . 5 |- ( A e. V -> ( A i^i ( Base ` W ) ) e. _V )
10	4, 9	syl 14	. . . 4 |- ( ph -> ( A i^i ( Base ` W ) ) e. _V )
11		resseqnbasd.f	. . . . 5 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
12		resseqnbas.n	. . . . 5 |- ( E ` ndx ) =/= ( Base ` ndx )
13		basendxnn 13201	. . . . 5 |- ( Base ` ndx ) e. NN
14	11, 12, 13	setsslnid 13197	. . . 4 |- ( ( W e. X /\ ( A i^i ( Base ` W ) ) e. _V ) -> ( E ` W ) = ( E ` ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
15	3, 10, 14	syl2anc 411	. . 3 |- ( ph -> ( E ` W ) = ( E ` ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
16	8, 15	eqtr4d 2267	. 2 |- ( ph -> ( E ` R ) = ( E ` W ) )
17	1, 16	eqtr4id 2283	1 |- ( ph -> C = ( E ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   =/= wne 2403   _Vcvv 2803   i^i cin 3200   <.cop 3676   ` cfv 5333  (class class class)co 6028   NNcn 9185   ndxcnx 13142   sSet csts 13143  Slot cslot 13144   Basecbs 13145   ↾_s cress 13146

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-inn 9186  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153

This theorem is referenced by:  ressplusgd  13275  ressmulrg  13291  ressscag  13329  ressvscag  13330  ressipg  13331