Theorem isstructr 13216

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 4816	. . . 4 |- ( M ( <_ i^i ( NN X. NN ) ) N <-> ( M e. NN /\ N e. NN /\ M <_ N ) )
2		df-br 4109	. . . 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 1030	. 2 |- ( ( ( M e. NN /\ N e. NN /\ M <_ N ) /\ ( Fun ( F \ { (/) } ) /\ F e. V /\ dom F C_ ( M ... N ) ) ) -> Fun ( F \ { (/) } ) )
6		simpr2 1031	. 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 6052	. . . . . 6 |- ( M ... N ) = ( ... ` <. M , N >. )
8	7	sseq2i 3264	. . . . 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 1047	. . 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 13212	. 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 1275	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 1005   e. wcel 2203   \ cdif 3207   i^i cin 3209   C_ wss 3210   (/)c0 3507   {csn 3688   <.cop 3691   class class class wbr 4108   X. cxp 4746   dom cdm 4748   Fun wfun 5345   ` cfv 5351  (class class class)co 6049   <_ cle 8305   NNcn 9233   ...cfz 10338   Struct cstr 13197

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-iota 5311  df-fun 5353  df-fv 5359  df-ov 6052  df-struct 13203

This theorem is referenced by:  strleund  13305  strleun  13306  strext  13307  strle1g  13308