Theorem isstruct2im 11969

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 11968	. . . 4 |- Rel Struct
2	1	brrelex12i 4581	. . 3 |- ( F Struct X -> ( F e. _V /\ X e. _V ) )
3		simpr 109	. . . . . 6 |- ( ( f = F /\ x = X ) -> x = X )
4	3	eleq1d 2208	. . . . 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 3193	. . . . . 6 |- ( ( f = F /\ x = X ) -> ( f \ { (/) } ) = ( F \ { (/) } ) )
7	6	funeqd 5145	. . . . 5 |- ( ( f = F /\ x = X ) -> ( Fun ( f \ { (/) } ) <-> Fun ( F \ { (/) } ) ) )
8	5	dmeqd 4741	. . . . . 6 |- ( ( f = F /\ x = X ) -> dom f = dom F )
9	3	fveq2d 5425	. . . . . 6 |- ( ( f = F /\ x = X ) -> ( ... ` x ) = ( ... ` X ) )
10	8, 9	sseq12d 3128	. . . . 5 |- ( ( f = F /\ x = X ) -> ( dom f C_ ( ... ` x ) <-> dom F C_ ( ... ` X ) ) )
11	4, 7, 10	3anbi123d 1290	. . . 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 11961	. . . 4 |- Struct = { <. f , x >. | ( x e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( f \ { (/) } ) /\ dom f C_ ( ... ` x ) ) }
13	11, 12	brabga 4186	. . 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 962   = wceq 1331   e. wcel 1480   _Vcvv 2686   \ cdif 3068   i^i cin 3070   C_ wss 3071   (/)c0 3363   {csn 3527   class class class wbr 3929   X. cxp 4537   dom cdm 4539   Fun wfun 5117   ` cfv 5123   <_ cle 7801   NNcn 8720   ...cfz 9790   Struct cstr 11955

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-struct 11961

This theorem is referenced by:  structn0fun  11972  isstructim  11973