Theorem resseqnbasd 13114

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 13105	. . . . . 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 2274	. . . 4 |- ( ph -> R = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
8	7	fveq2d 5633	. . 3 |- ( ph -> ( E ` R ) = ( E ` ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
9		inex1g 4220	. . . . 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 13096	. . . . 5 |- ( Base ` ndx ) e. NN
14	11, 12, 13	setsslnid 13092	. . . 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 2265	. 2 |- ( ph -> ( E ` R ) = ( E ` W ) )
17	1, 16	eqtr4id 2281	1 |- ( ph -> C = ( E ` R ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   =/= wne 2400   _Vcvv 2799   i^i cin 3196   <.cop 3669   ` cfv 5318  (class class class)co 6007   NNcn 9118   ndxcnx 13037   sSet csts 13038  Slot cslot 13039   Basecbs 13040   ↾_s cress 13041

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1re 8101  ax-addrcl 8104

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-inn 9119  df-ndx 13043  df-slot 13044  df-base 13046  df-sets 13047  df-iress 13048

This theorem is referenced by:  ressplusgd  13170  ressmulrg  13186  ressscag  13224  ressvscag  13225  ressipg  13226