Theorem isstruct2 15867

Description: The property of being a structure with components in ( 1st ` X ) ... ( 2nd ` X ). (Contributed by Mario Carneiro, 29-Aug-2015.)

Assertion

Ref	Expression
isstruct2	|- ( F Struct X <-> ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) )

Proof of Theorem isstruct2

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

Step	Hyp	Ref	Expression
1		brstruct 15866	. . 3 |- Rel Struct
2		brrelex12 5155	. . 3 |- ( ( Rel Struct /\ F Struct X ) -> ( F e. _V /\ X e. _V ) )
3	1, 2	mpan 706	. 2 |- ( F Struct X -> ( F e. _V /\ X e. _V ) )
4		ssun1 3776	. . . . 5 |- F C_ ( F u. { (/) } )
5		undif1 4043	. . . . 5 |- ( ( F \ { (/) } ) u. { (/) } ) = ( F u. { (/) } )
6	4, 5	sseqtr4i 3638	. . . 4 |- F C_ ( ( F \ { (/) } ) u. { (/) } )
7		simp2 1062	. . . . . . 7 |- ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) -> Fun ( F \ { (/) } ) )
8		funfn 5918	. . . . . . 7 |- ( Fun ( F \ { (/) } ) <-> ( F \ { (/) } ) Fn dom ( F \ { (/) } ) )
9	7, 8	sylib 208	. . . . . 6 |- ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) -> ( F \ { (/) } ) Fn dom ( F \ { (/) } ) )
10		inss2 3834	. . . . . . . . . . . 12 |- ( <_ i^i ( NN X. NN ) ) C_ ( NN X. NN )
11	10	sseli 3599	. . . . . . . . . . 11 |- ( X e. ( <_ i^i ( NN X. NN ) ) -> X e. ( NN X. NN ) )
12		1st2nd2 7205	. . . . . . . . . . 11 |- ( X e. ( NN X. NN ) -> X = <. ( 1st ` X ) , ( 2nd ` X ) >. )
13	11, 12	syl 17	. . . . . . . . . 10 |- ( X e. ( <_ i^i ( NN X. NN ) ) -> X = <. ( 1st ` X ) , ( 2nd ` X ) >. )
14	13	3ad2ant1 1082	. . . . . . . . 9 |- ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) -> X = <. ( 1st ` X ) , ( 2nd ` X ) >. )
15	14	fveq2d 6195	. . . . . . . 8 |- ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) -> ( ... ` X ) = ( ... ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) )
16		df-ov 6653	. . . . . . . . 9 |- ( ( 1st ` X ) ... ( 2nd ` X ) ) = ( ... ` <. ( 1st ` X ) , ( 2nd ` X ) >. )
17		fzfi 12771	. . . . . . . . 9 |- ( ( 1st ` X ) ... ( 2nd ` X ) ) e. Fin
18	16, 17	eqeltrri 2698	. . . . . . . 8 |- ( ... ` <. ( 1st ` X ) , ( 2nd ` X ) >. ) e. Fin
19	15, 18	syl6eqel 2709	. . . . . . 7 |- ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) -> ( ... ` X ) e. Fin )
20		difss 3737	. . . . . . . . 9 |- ( F \ { (/) } ) C_ F
21		dmss 5323	. . . . . . . . 9 |- ( ( F \ { (/) } ) C_ F -> dom ( F \ { (/) } ) C_ dom F )
22	20, 21	ax-mp 5	. . . . . . . 8 |- dom ( F \ { (/) } ) C_ dom F
23		simp3 1063	. . . . . . . 8 |- ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) -> dom F C_ ( ... ` X ) )
24	22, 23	syl5ss 3614	. . . . . . 7 |- ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) -> dom ( F \ { (/) } ) C_ ( ... ` X ) )
25		ssfi 8180	. . . . . . 7 |- ( ( ( ... ` X ) e. Fin /\ dom ( F \ { (/) } ) C_ ( ... ` X ) ) -> dom ( F \ { (/) } ) e. Fin )
26	19, 24, 25	syl2anc 693	. . . . . 6 |- ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) -> dom ( F \ { (/) } ) e. Fin )
27		fnfi 8238	. . . . . 6 |- ( ( ( F \ { (/) } ) Fn dom ( F \ { (/) } ) /\ dom ( F \ { (/) } ) e. Fin ) -> ( F \ { (/) } ) e. Fin )
28	9, 26, 27	syl2anc 693	. . . . 5 |- ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) -> ( F \ { (/) } ) e. Fin )
29		p0ex 4853	. . . . 5 |- { (/) } e. _V
30		unexg 6959	. . . . 5 |- ( ( ( F \ { (/) } ) e. Fin /\ { (/) } e. _V ) -> ( ( F \ { (/) } ) u. { (/) } ) e. _V )
31	28, 29, 30	sylancl 694	. . . 4 |- ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) -> ( ( F \ { (/) } ) u. { (/) } ) e. _V )
32		ssexg 4804	. . . 4 |- ( ( F C_ ( ( F \ { (/) } ) u. { (/) } ) /\ ( ( F \ { (/) } ) u. { (/) } ) e. _V ) -> F e. _V )
33	6, 31, 32	sylancr 695	. . 3 |- ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) -> F e. _V )
34		elex 3212	. . . 4 |- ( X e. ( <_ i^i ( NN X. NN ) ) -> X e. _V )
35	34	3ad2ant1 1082	. . 3 |- ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) -> X e. _V )
36	33, 35	jca 554	. 2 |- ( ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) -> ( F e. _V /\ X e. _V ) )
37		simpr 477	. . . . 5 |- ( ( f = F /\ x = X ) -> x = X )
38	37	eleq1d 2686	. . . 4 |- ( ( f = F /\ x = X ) -> ( x e. ( <_ i^i ( NN X. NN ) ) <-> X e. ( <_ i^i ( NN X. NN ) ) ) )
39		simpl 473	. . . . . 6 |- ( ( f = F /\ x = X ) -> f = F )
40	39	difeq1d 3727	. . . . 5 |- ( ( f = F /\ x = X ) -> ( f \ { (/) } ) = ( F \ { (/) } ) )
41	40	funeqd 5910	. . . 4 |- ( ( f = F /\ x = X ) -> ( Fun ( f \ { (/) } ) <-> Fun ( F \ { (/) } ) ) )
42	39	dmeqd 5326	. . . . 5 |- ( ( f = F /\ x = X ) -> dom f = dom F )
43	37	fveq2d 6195	. . . . 5 |- ( ( f = F /\ x = X ) -> ( ... ` x ) = ( ... ` X ) )
44	42, 43	sseq12d 3634	. . . 4 |- ( ( f = F /\ x = X ) -> ( dom f C_ ( ... ` x ) <-> dom F C_ ( ... ` X ) ) )
45	38, 41, 44	3anbi123d 1399	. . 3 |- ( ( 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 ) ) ) )
46		df-struct 15859	. . 3 |- Struct = { <. f , x >. | ( x e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( f \ { (/) } ) /\ dom f C_ ( ... ` x ) ) }
47	45, 46	brabga 4989	. 2 |- ( ( F e. _V /\ X e. _V ) -> ( F Struct X <-> ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) ) )
48	3, 36, 47	pm5.21nii 368	1 |- ( F Struct X <-> ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   class class class wbr 4653   X. cxp 5112   dom cdm 5114   Rel wrel 5119   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Fincfn 7955   <_ cle 10075   NNcn 11020   ...cfz 12326   Struct cstr 15853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859

This theorem is referenced by:  structn0fun  15869  isstruct  15870  setsstruct2  15896