Theorem isstruct2r 13365

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 531	. 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 533	. 2 |- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> Fun ( F \ { (/) } ) )
3		simprr 537	. 2 |- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> dom F C_ ( ... ` X ) )
4		simprl 535	. . . 4 |- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> F e. V )
5	4	elexd 2835	. . 3 |- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> F e. _V )
6		elex 2833	. . . 4 |- ( X e. ( <_ i^i ( NN X. NN ) ) -> X e. _V )
7	6	ad2antrr 492	. . 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 2307	. . . . 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 3346	. . . . . 6 |- ( ( f = F /\ x = X ) -> ( f \ { (/) } ) = ( F \ { (/) } ) )
12	11	funeqd 5399	. . . . 5 |- ( ( f = F /\ x = X ) -> ( Fun ( f \ { (/) } ) <-> Fun ( F \ { (/) } ) ) )
13	10	dmeqd 4983	. . . . . 6 |- ( ( f = F /\ x = X ) -> dom f = dom F )
14	8	fveq2d 5699	. . . . . 6 |- ( ( f = F /\ x = X ) -> ( ... ` x ) = ( ... ` X ) )
15	13, 14	sseq12d 3279	. . . . 5 |- ( ( f = F /\ x = X ) -> ( dom f C_ ( ... ` x ) <-> dom F C_ ( ... ` X ) ) )
16	9, 12, 15	3anbi123d 1353	. . . 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 13356	. . . 4 |- Struct = { <. f , x >. | ( x e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( f \ { (/) } ) /\ dom f C_ ( ... ` x ) ) }
18	16, 17	brabga 4406	. . 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 415	. 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 1211	1 |- ( ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) ) /\ ( F e. V /\ dom F C_ ( ... ` X ) ) ) -> F Struct X )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   \ cdif 3217   i^i cin 3219   C_ wss 3220   (/)c0 3520   {csn 3709   class class class wbr 4130   X. cxp 4772   dom cdm 4774   Fun wfun 5371   ` cfv 5377   <_ cle 8361   NNcn 9305   ...cfz 10413   Struct cstr 13350

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-struct 13356

This theorem is used by:  isstructr  13369