Theorem isstruct2im 12464

Description: The property of being a structure with components in ( 1st ` X ) ... ( 2nd ` X ). (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)

Assertion

Ref	Expression
isstruct2im	|- ( F Struct X -> ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) )

Proof of Theorem isstruct2im

Dummy variables x f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brstruct 12463	. . . 4 |- Rel Struct
2	1	brrelex12i 4667	. . 3 |- ( F Struct X -> ( F e. _V /\ X e. _V ) )
3		simpr 110	. . . . . 6 |- ( ( f = F /\ x = X ) -> x = X )
4	3	eleq1d 2246	. . . . 5 |- ( ( f = F /\ x = X ) -> ( x e. ( <_ i^i ( NN X. NN ) ) <-> X e. ( <_ i^i ( NN X. NN ) ) ) )
5		simpl 109	. . . . . . 7 |- ( ( f = F /\ x = X ) -> f = F )
6	5	difeq1d 3252	. . . . . 6 |- ( ( f = F /\ x = X ) -> ( f \ { (/) } ) = ( F \ { (/) } ) )
7	6	funeqd 5237	. . . . 5 |- ( ( f = F /\ x = X ) -> ( Fun ( f \ { (/) } ) <-> Fun ( F \ { (/) } ) ) )
8	5	dmeqd 4828	. . . . . 6 |- ( ( f = F /\ x = X ) -> dom f = dom F )
9	3	fveq2d 5518	. . . . . 6 |- ( ( f = F /\ x = X ) -> ( ... ` x ) = ( ... ` X ) )
10	8, 9	sseq12d 3186	. . . . 5 |- ( ( f = F /\ x = X ) -> ( dom f C_ ( ... ` x ) <-> dom F C_ ( ... ` X ) ) )
11	4, 7, 10	3anbi123d 1312	. . . 4 |- ( ( f = F /\ x = X ) -> ( ( x e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( f \ { (/) } ) /\ dom f C_ ( ... ` x ) ) <-> ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) ) )
12		df-struct 12456	. . . 4 |- Struct = { <. f , x >. | ( x e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( f \ { (/) } ) /\ dom f C_ ( ... ` x ) ) }
13	11, 12	brabga 4263	. . 3 |- ( ( F e. _V /\ X e. _V ) -> ( F Struct X <-> ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) ) )
14	2, 13	syl 14	. 2 |- ( F Struct X -> ( F Struct X <-> ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) ) )
15	14	ibi 176	1 |- ( F Struct X -> ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   _Vcvv 2737   \ cdif 3126   i^i cin 3128   C_ wss 3129   (/)c0 3422   {csn 3592   class class class wbr 4002   X. cxp 4623   dom cdm 4625   Fun wfun 5209   ` cfv 5215   <_ cle 7989   NNcn 8915   ...cfz 10004   Struct cstr 12450

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-iota 5177  df-fun 5217  df-fv 5223  df-struct 12456

This theorem is referenced by:  structn0fun  12467  isstructim  12468