Theorem isstructr 13099

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 4793	. . . 4 |- ( M ( <_ i^i ( NN X. NN ) ) N <-> ( M e. NN /\ N e. NN /\ M <_ N ) )
2		df-br 4089	. . . 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 1029	. 2 |- ( ( ( M e. NN /\ N e. NN /\ M <_ N ) /\ ( Fun ( F \ { (/) } ) /\ F e. V /\ dom F C_ ( M ... N ) ) ) -> Fun ( F \ { (/) } ) )
6		simpr2 1030	. 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 6021	. . . . . 6 |- ( M ... N ) = ( ... ` <. M , N >. )
8	7	sseq2i 3254	. . . . 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 1046	. . 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 13095	. 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 1274	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 1004   e. wcel 2202   \ cdif 3197   i^i cin 3199   C_ wss 3200   (/)c0 3494   {csn 3669   <.cop 3672   class class class wbr 4088   X. cxp 4723   dom cdm 4725   Fun wfun 5320   ` cfv 5326  (class class class)co 6018   <_ cle 8215   NNcn 9143   ...cfz 10243   Struct cstr 13080

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6021  df-struct 13086

This theorem is referenced by:  strleund  13188  strleun  13189  strext  13190  strle1g  13191