Theorem setsslnid 13382

Description: Value of the structure replacement function at an untouched index. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2023.)

Hypotheses

Ref	Expression
setsslid.e	|- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
setsslnid.n	|- ( E ` ndx ) =/= D
setsslnid.d	|- D e. NN

Assertion

Ref	Expression
setsslnid	|- ( ( W e. A /\ C e. V ) -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )

Proof of Theorem setsslnid

Step	Hyp	Ref	Expression
1		setsslnid.d	. . . . 5 |- D e. NN
2		setsresg 13368	. . . . 5 |- ( ( W e. A /\ D e. NN /\ C e. V ) -> ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) = ( W |` ( _V \ { D } ) ) )
3	1, 2	mp3an2 1366	. . . 4 |- ( ( W e. A /\ C e. V ) -> ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) = ( W |` ( _V \ { D } ) ) )
4	3	fveq1d 5692	. . 3 |- ( ( W e. A /\ C e. V ) -> ( ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( ( W |` ( _V \ { D } ) ) ` ( E ` ndx ) ) )
5		setsslid.e	. . . . . . 7 |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )
6	5	simpri 113	. . . . . 6 |- ( E ` ndx ) e. NN
7	6	elexi 2834	. . . . 5 |- ( E ` ndx ) e. _V
8		setsslnid.n	. . . . 5 |- ( E ` ndx ) =/= D
9		eldifsn 3836	. . . . 5 |- ( ( E ` ndx ) e. ( _V \ { D } ) <-> ( ( E ` ndx ) e. _V /\ ( E ` ndx ) =/= D ) )
10	7, 8, 9	mpbir2an 955	. . . 4 |- ( E ` ndx ) e. ( _V \ { D } )
11		fvres 5714	. . . 4 |- ( ( E ` ndx ) e. ( _V \ { D } ) -> ( ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( ( W sSet <. D , C >. ) ` ( E ` ndx ) ) )
12	10, 11	ax-mp 5	. . 3 |- ( ( ( W sSet <. D , C >. ) |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( ( W sSet <. D , C >. ) ` ( E ` ndx ) )
13		fvres 5714	. . . 4 |- ( ( E ` ndx ) e. ( _V \ { D } ) -> ( ( W |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( W ` ( E ` ndx ) ) )
14	10, 13	ax-mp 5	. . 3 |- ( ( W |` ( _V \ { D } ) ) ` ( E ` ndx ) ) = ( W ` ( E ` ndx ) )
15	4, 12, 14	3eqtr3g 2294	. 2 |- ( ( W e. A /\ C e. V ) -> ( ( W sSet <. D , C >. ) ` ( E ` ndx ) ) = ( W ` ( E ` ndx ) ) )
16	5	simpli 111	. . 3 |- E = Slot ( E ` ndx )
17		setsex 13362	. . . 4 |- ( ( W e. A /\ D e. NN /\ C e. V ) -> ( W sSet <. D , C >. ) e. _V )
18	1, 17	mp3an2 1366	. . 3 |- ( ( W e. A /\ C e. V ) -> ( W sSet <. D , C >. ) e. _V )
19	6	a1i 9	. . 3 |- ( ( W e. A /\ C e. V ) -> ( E ` ndx ) e. NN )
20	16, 18, 19	strnfvnd 13350	. 2 |- ( ( W e. A /\ C e. V ) -> ( E ` ( W sSet <. D , C >. ) ) = ( ( W sSet <. D , C >. ) ` ( E ` ndx ) ) )
21		simpl 109	. . 3 |- ( ( W e. A /\ C e. V ) -> W e. A )
22	16, 21, 19	strnfvnd 13350	. 2 |- ( ( W e. A /\ C e. V ) -> ( E ` W ) = ( W ` ( E ` ndx ) ) )
23	15, 20, 22	3eqtr4rd 2282	1 |- ( ( W e. A /\ C e. V ) -> ( E ` W ) = ( E ` ( W sSet <. D , C >. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420   _Vcvv 2821   \ cdif 3217   {csn 3705   <.cop 3708   |` cres 4771   ` cfv 5372  (class class class)co 6075   NNcn 9283   ndxcnx 13327   sSet csts 13328  Slot cslot 13329

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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-slot 13334  df-sets 13337

This theorem is referenced by:  resseqnbasd  13404  mgpbasg  14200  mgpscag  14201  mgptsetg  14202  mgpdsg  14204  opprsllem  14352  rmodislmod  14660  sralemg  14747  srascag  14751  sravscag  14752  zlmlemg  14935  zlmsca  14939  znbaslemnn  14946  setsmsbasg  15503  setsmsdsg  15504  setsvtx  16206