Theorem resseqnbasd 13427

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 13418	. . . . . 6 |- ( ( W e. X /\ A e. V ) -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
6	3, 4, 5	syl2anc 415	. . . . 5 |- ( ph -> ( W ↾s A ) = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
7	2, 6	eqtrid 2283	. . . 4 |- ( ph -> R = ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) )
8	7	fveq2d 5699	. . 3 |- ( ph -> ( E ` R ) = ( E ` ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
9		inex1g 4269	. . . . 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 13408	. . . . 5 |- ( Base ` ndx ) e. NN
14	11, 12, 13	setsslnid 13404	. . . 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 415	. . 3 |- ( ph -> ( E ` W ) = ( E ` ( W sSet <. ( Base ` ndx ) , ( A i^i ( Base ` W ) ) >. ) ) )
16	8, 15	eqtr4d 2274	. 2 |- ( ph -> ( E ` R ) = ( E ` W ) )
17	1, 16	eqtr4id 2290	1 |- ( ph -> C = ( E ` R ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420   _Vcvv 2821   i^i cin 3219   <.cop 3712   ` cfv 5377  (class class class)co 6085   NNcn 9304   ndxcnx 13349   sSet csts 13350  Slot cslot 13351   Basecbs 13352   ↾_s cress 13353

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-inn 9305  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-iress 13360

This theorem is used by:  ressplusgd  13483  ressmulrg  13499  ressscag  13537  ressvscag  13538  ressipg  13539