Theorem isstructr 12469

Description: The property of being a structure with components in M ... N. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 18-Jan-2023.)

Assertion

Ref	Expression
isstructr	|- ( ( ( M e. NN /\ N e. NN /\ M <_ N ) /\ ( Fun ( F \ { (/) } ) /\ F e. V /\ dom F C_ ( M ... N ) ) ) -> F Struct <. M , N >. )

Proof of Theorem isstructr

Step	Hyp	Ref	Expression
1		brinxp2 4692	. . . 4 |- ( M ( <_ i^i ( NN X. NN ) ) N <-> ( M e. NN /\ N e. NN /\ M <_ N ) )
2		df-br 4003	. . . 4 |- ( M ( <_ i^i ( NN X. NN ) ) N <-> <. M , N >. e. ( <_ i^i ( NN X. NN ) ) )
3	1, 2	sylbb1 137	. . 3 |- ( ( M e. NN /\ N e. NN /\ M <_ N ) -> <. M , N >. e. ( <_ i^i ( NN X. NN ) ) )
4	3	adantr 276	. 2 |- ( ( ( M e. NN /\ N e. NN /\ M <_ N ) /\ ( Fun ( F \ { (/) } ) /\ F e. V /\ dom F C_ ( M ... N ) ) ) -> <. M , N >. e. ( <_ i^i ( NN X. NN ) ) )
5		simpr1 1003	. 2 |- ( ( ( M e. NN /\ N e. NN /\ M <_ N ) /\ ( Fun ( F \ { (/) } ) /\ F e. V /\ dom F C_ ( M ... N ) ) ) -> Fun ( F \ { (/) } ) )
6		simpr2 1004	. 2 |- ( ( ( M e. NN /\ N e. NN /\ M <_ N ) /\ ( Fun ( F \ { (/) } ) /\ F e. V /\ dom F C_ ( M ... N ) ) ) -> F e. V )
7		df-ov 5875	. . . . . 6 |- ( M ... N ) = ( ... ` <. M , N >. )
8	7	sseq2i 3182	. . . . 5 |- ( dom F C_ ( M ... N ) <-> dom F C_ ( ... ` <. M , N >. ) )
9	8	biimpi 120	. . . 4 |- ( dom F C_ ( M ... N ) -> dom F C_ ( ... ` <. M , N >. ) )
10	9	3ad2ant3 1020	. . 3 |- ( ( Fun ( F \ { (/) } ) /\ F e. V /\ dom F C_ ( M ... N ) ) -> dom F C_ ( ... ` <. M , N >. ) )
11	10	adantl 277	. 2 |- ( ( ( M e. NN /\ N e. NN /\ M <_ N ) /\ ( Fun ( F \ { (/) } ) /\ F e. V /\ dom F C_ ( M ... N ) ) ) -> dom F C_ ( ... ` <. M , N >. ) )
12		isstruct2r 12465	. 2 |- ( ( ( <. M , N >. e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` <. M , N >. ) ) ) -> F Struct <. M , N >. )
13	4, 5, 6, 11, 12	syl22anc 1239	1 |- ( ( ( M e. NN /\ N e. NN /\ M <_ N ) /\ ( Fun ( F \ { (/) } ) /\ F e. V /\ dom F C_ ( M ... N ) ) ) -> F Struct <. M , N >. )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   e. wcel 2148   \ cdif 3126   i^i cin 3128   C_ wss 3129   (/)c0 3422   {csn 3592   <.cop 3595   class class class wbr 4002   X. cxp 4623   dom cdm 4625   Fun wfun 5209   ` cfv 5215  (class class class)co 5872   <_ 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-ov 5875  df-struct 12456

This theorem is referenced by:  strleund  12554  strleun  12555  strext  12556  strle1g  12557