Theorem isstruct2r 13116

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
isstruct2r	|- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> F Struct X )

Proof of Theorem isstruct2r

Dummy variables x f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 527	. 2 |- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> X e. ( <_ i^i ( NN X. NN ) ) )
2		simplr 529	. 2 |- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> Fun ( F \ { (/) } ) )
3		simprr 533	. 2 |- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> dom F C_ ( ... ` X ) )
4		simprl 531	. . . 4 |- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> F e. V )
5	4	elexd 2815	. . 3 |- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> F e. _V )
6		elex 2813	. . . 4 |- ( X e. ( <_ i^i ( NN X. NN ) ) -> X e. _V )
7	6	ad2antrr 488	. . 3 |- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> X e. _V )
8		simpr 110	. . . . . 6 |- ( ( f = F /\ x = X ) -> x = X )
9	8	eleq1d 2299	. . . . 5 |- ( ( f = F /\ x = X ) -> ( x e. ( <_ i^i ( NN X. NN ) ) <-> X e. ( <_ i^i ( NN X. NN ) ) ) )
10		simpl 109	. . . . . . 7 |- ( ( f = F /\ x = X ) -> f = F )
11	10	difeq1d 3323	. . . . . 6 |- ( ( f = F /\ x = X ) -> ( f \ { (/) } ) = ( F \ { (/) } ) )
12	11	funeqd 5350	. . . . 5 |- ( ( f = F /\ x = X ) -> ( Fun ( f \ { (/) } ) <-> Fun ( F \ { (/) } ) ) )
13	10	dmeqd 4935	. . . . . 6 |- ( ( f = F /\ x = X ) -> dom f = dom F )
14	8	fveq2d 5646	. . . . . 6 |- ( ( f = F /\ x = X ) -> ( ... ` x ) = ( ... ` X ) )
15	13, 14	sseq12d 3257	. . . . 5 |- ( ( f = F /\ x = X ) -> ( dom f C_ ( ... ` x ) <-> dom F C_ ( ... ` X ) ) )
16	9, 12, 15	3anbi123d 1348	. . . 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 ) ) ) )
17		df-struct 13107	. . . 4 |- Struct = { <. f , x >. | ( x e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( f \ { (/) } ) /\ dom f C_ ( ... ` x ) ) }
18	16, 17	brabga 4360	. . 3 |- ( ( F e. _V /\ X e. _V ) -> ( F Struct X <-> ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) ) )
19	5, 7, 18	syl2anc 411	. 2 |- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> ( F Struct X <-> ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) ) )
20	1, 2, 3, 19	mpbir3and 1206	1 |- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> F Struct X )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2201   _Vcvv 2801   \ cdif 3196   i^i cin 3198   C_ wss 3199   (/)c0 3493   {csn 3670   class class class wbr 4089   X. cxp 4725   dom cdm 4727   Fun wfun 5322   ` cfv 5328   <_ cle 8220   NNcn 9148   ...cfz 10248   Struct cstr 13101

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 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-rex 2515  df-rab 2518  df-v 2803  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-iota 5288  df-fun 5330  df-fv 5336  df-struct 13107

This theorem is referenced by:  isstructr  13120