Theorem isstruct2im 12008

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 12007	. . . 4 |- Rel Struct
2	1	brrelex12i 4589	. . 3 |- ( F Struct X -> ( F e. _V /\ X e. _V ) )
3		simpr 109	. . . . . 6 |- ( ( f = F /\ x = X ) -> x = X )
4	3	eleq1d 2209	. . . . 5 |- ( ( f = F /\ x = X ) -> ( x e. ( <_ i^i ( NN X. NN ) ) <-> X e. ( <_ i^i ( NN X. NN ) ) ) )
5		simpl 108	. . . . . . 7 |- ( ( f = F /\ x = X ) -> f = F )
6	5	difeq1d 3198	. . . . . 6 |- ( ( f = F /\ x = X ) -> ( f \ { (/) } ) = ( F \ { (/) } ) )
7	6	funeqd 5153	. . . . 5 |- ( ( f = F /\ x = X ) -> ( Fun ( f \ { (/) } ) <-> Fun ( F \ { (/) } ) ) )
8	5	dmeqd 4749	. . . . . 6 |- ( ( f = F /\ x = X ) -> dom f = dom F )
9	3	fveq2d 5433	. . . . . 6 |- ( ( f = F /\ x = X ) -> ( ... ` x ) = ( ... ` X ) )
10	8, 9	sseq12d 3133	. . . . 5 |- ( ( f = F /\ x = X ) -> ( dom f C_ ( ... ` x ) <-> dom F C_ ( ... ` X ) ) )
11	4, 7, 10	3anbi123d 1291	. . . 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 12000	. . . 4 |- Struct = { <. f , x >. | ( x e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( f \ { (/) } ) /\ dom f C_ ( ... ` x ) ) }
13	11, 12	brabga 4194	. . 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 175	1 |- ( F Struct X -> ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1332   e. wcel 1481   _Vcvv 2689   \ cdif 3073   i^i cin 3075   C_ wss 3076   (/)c0 3368   {csn 3532   class class class wbr 3937   X. cxp 4545   dom cdm 4547   Fun wfun 5125   ` cfv 5131   <_ cle 7825   NNcn 8744   ...cfz 9821   Struct cstr 11994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-struct 12000

This theorem is referenced by:  structn0fun  12011  isstructim  12012