Theorem strndxid 12455

Description: The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.)

Hypotheses

Ref	Expression
strndxid.s	|- ( ph -> S e. V )
strndxid.e	|- E = Slot N
strndxid.n	|- N e. NN

Assertion

Ref	Expression
strndxid	|- ( ph -> ( S ` ( E ` ndx ) ) = ( E ` S ) )

Proof of Theorem strndxid

Step	Hyp	Ref	Expression
1		strndxid.e	. . . 4 |- E = Slot N
2		strndxid.n	. . . 4 |- N e. NN
3	1, 2	ndxid 12451	. . 3 |- E = Slot ( E ` ndx )
4		strndxid.s	. . 3 |- ( ph -> S e. V )
5	1, 2	ndxarg 12450	. . . . 5 |- ( E ` ndx ) = N
6	5, 2	eqeltri 2248	. . . 4 |- ( E ` ndx ) e. NN
7	6	a1i 9	. . 3 |- ( ph -> ( E ` ndx ) e. NN )
8	3, 4, 7	strnfvnd 12447	. 2 |- ( ph -> ( E ` S ) = ( S ` ( E ` ndx ) ) )
9	8	eqcomd 2181	1 |- ( ph -> ( S ` ( E ` ndx ) ) = ( E ` S ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2146   ` cfv 5208   NNcn 8890   ndxcnx 12424  Slot cslot 12426

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-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fv 5216  df-inn 8891  df-ndx 12430  df-slot 12431

This theorem is referenced by: (None)