Theorem resseqnbasd 13286

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 13277	. . . . . 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 2277	. . . 4 |- ( ph -> R = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
8	7	fveq2d 5674	. . 3 |- ( ph -> ( E ` R ) = ( E ` ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
9		inex1g 4246	. . . . 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 13268	. . . . 5 |- ( Base ` ndx ) e. NN
14	11, 12, 13	setsslnid 13264	. . . 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 2268	. 2 |- ( ph -> ( E ` R ) = ( E ` W ) )
17	1, 16	eqtr4id 2284	1 |- ( ph -> C = ( E ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   =/= wne 2412   _Vcvv 2813   i^i cin 3210   <.cop 3692   ` cfv 5352  (class class class)co 6050   NNcn 9237   ndxcnx 13209   sSet csts 13210  Slot cslot 13211   Basecbs 13212   ↾_s cress 13213

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-iress 13220

This theorem is referenced by:  ressplusgd  13342  ressmulrg  13358  ressscag  13396  ressvscag  13397  ressipg  13398