Theorem dfrtrcl2 13802

Description: The two definitions t* and t*rec of the reflexive, transitive closure coincide if R is indeed a relation. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)

Hypotheses

Ref	Expression
drrtrcl2.1	|- ( ph -> Rel R )
drrtrcl2.2	|- ( ph -> R e. _V )

Assertion

Ref	Expression
dfrtrcl2	|- ( ph -> ( t* ` R ) = ( t*rec ` R ) )

Proof of Theorem dfrtrcl2

Dummy variables x z s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2623	. . . 4 |- ( ph -> ( x e. _V |-> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } ) = ( x e. _V |-> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } ) )
2		dmeq 5324	. . . . . . . . . . 11 |- ( x = R -> dom x = dom R )
3		rneq 5351	. . . . . . . . . . 11 |- ( x = R -> ran x = ran R )
4	2, 3	uneq12d 3768	. . . . . . . . . 10 |- ( x = R -> ( dom x u. ran x ) = ( dom R u. ran R ) )
5	4	reseq2d 5396	. . . . . . . . 9 |- ( x = R -> ( _I |` ( dom x u. ran x ) ) = ( _I |` ( dom R u. ran R ) ) )
6	5	sseq1d 3632	. . . . . . . 8 |- ( x = R -> ( ( _I |` ( dom x u. ran x ) ) C_ z <-> ( _I |` ( dom R u. ran R ) ) C_ z ) )
7		id 22	. . . . . . . . 9 |- ( x = R -> x = R )
8	7	sseq1d 3632	. . . . . . . 8 |- ( x = R -> ( x C_ z <-> R C_ z ) )
9	6, 8	3anbi12d 1400	. . . . . . 7 |- ( x = R -> ( ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) <-> ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) ) )
10	9	abbidv 2741	. . . . . 6 |- ( x = R -> { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } = { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } )
11	10	inteqd 4480	. . . . 5 |- ( x = R -> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } = |^| { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } )
12	11	adantl 482	. . . 4 |- ( ( ph /\ x = R ) -> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } = |^| { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } )
13		drrtrcl2.2	. . . 4 |- ( ph -> R e. _V )
14		drrtrcl2.1	. . . . . . . . . 10 |- ( ph -> Rel R )
15		relfld 5661	. . . . . . . . . 10 |- ( Rel R -> U. U. R = ( dom R u. ran R ) )
16	14, 15	syl 17	. . . . . . . . 9 |- ( ph -> U. U. R = ( dom R u. ran R ) )
17	16	eqcomd 2628	. . . . . . . 8 |- ( ph -> ( dom R u. ran R ) = U. U. R )
18	14, 13	rtrclreclem1 13798	. . . . . . . . 9 |- ( ph -> ( _I |` U. U. R ) C_ ( t*rec ` R ) )
19		id 22	. . . . . . . . . . 11 |- ( ( dom R u. ran R ) = U. U. R -> ( dom R u. ran R ) = U. U. R )
20	19	reseq2d 5396	. . . . . . . . . 10 |- ( ( dom R u. ran R ) = U. U. R -> ( _I |` ( dom R u. ran R ) ) = ( _I |` U. U. R ) )
21	20	sseq1d 3632	. . . . . . . . 9 |- ( ( dom R u. ran R ) = U. U. R -> ( ( _I |` ( dom R u. ran R ) ) C_ ( t*rec ` R ) <-> ( _I |` U. U. R ) C_ ( t*rec ` R ) ) )
22	18, 21	syl5ibr 236	. . . . . . . 8 |- ( ( dom R u. ran R ) = U. U. R -> ( ph -> ( _I |` ( dom R u. ran R ) ) C_ ( t*rec ` R ) ) )
23	17, 22	mpcom 38	. . . . . . 7 |- ( ph -> ( _I |` ( dom R u. ran R ) ) C_ ( t*rec ` R ) )
24	13	rtrclreclem2 13799	. . . . . . 7 |- ( ph -> R C_ ( t*rec ` R ) )
25	14, 13	rtrclreclem3 13800	. . . . . . 7 |- ( ph -> ( ( t*rec ` R ) o. ( t*rec ` R ) ) C_ ( t*rec ` R ) )
26		fvex 6201	. . . . . . . 8 |- ( t*rec ` R ) e. _V
27		sseq2 3627	. . . . . . . . . . 11 |- ( z = ( t*rec ` R ) -> ( ( _I |` ( dom R u. ran R ) ) C_ z <-> ( _I |` ( dom R u. ran R ) ) C_ ( t*rec ` R ) ) )
28		sseq2 3627	. . . . . . . . . . 11 |- ( z = ( t*rec ` R ) -> ( R C_ z <-> R C_ ( t*rec ` R ) ) )
29		id 22	. . . . . . . . . . . . 13 |- ( z = ( t*rec ` R ) -> z = ( t*rec ` R ) )
30	29, 29	coeq12d 5286	. . . . . . . . . . . 12 |- ( z = ( t*rec ` R ) -> ( z o. z ) = ( ( t*rec ` R ) o. ( t*rec ` R ) ) )
31	30, 29	sseq12d 3634	. . . . . . . . . . 11 |- ( z = ( t*rec ` R ) -> ( ( z o. z ) C_ z <-> ( ( t*rec ` R ) o. ( t*rec ` R ) ) C_ ( t*rec ` R ) ) )
32	27, 28, 31	3anbi123d 1399	. . . . . . . . . 10 |- ( z = ( t*rec ` R ) -> ( ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) <-> ( ( _I |` ( dom R u. ran R ) ) C_ ( t*rec ` R ) /\ R C_ ( t*rec ` R ) /\ ( ( t*rec ` R ) o. ( t*rec ` R ) ) C_ ( t*rec ` R ) ) ) )
33	32	a1i 11	. . . . . . . . 9 |- ( ph -> ( z = ( t*rec ` R ) -> ( ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) <-> ( ( _I |` ( dom R u. ran R ) ) C_ ( t*rec ` R ) /\ R C_ ( t*rec ` R ) /\ ( ( t*rec ` R ) o. ( t*rec ` R ) ) C_ ( t*rec ` R ) ) ) ) )
34	33	alrimiv 1855	. . . . . . . 8 |- ( ph -> A. z ( z = ( t*rec ` R ) -> ( ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) <-> ( ( _I |` ( dom R u. ran R ) ) C_ ( t*rec ` R ) /\ R C_ ( t*rec ` R ) /\ ( ( t*rec ` R ) o. ( t*rec ` R ) ) C_ ( t*rec ` R ) ) ) ) )
35		elabgt 3347	. . . . . . . 8 |- ( ( ( t*rec ` R ) e. _V /\ A. z ( z = ( t*rec ` R ) -> ( ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) <-> ( ( _I |` ( dom R u. ran R ) ) C_ ( t*rec ` R ) /\ R C_ ( t*rec ` R ) /\ ( ( t*rec ` R ) o. ( t*rec ` R ) ) C_ ( t*rec ` R ) ) ) ) ) -> ( ( t*rec ` R ) e. { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } <-> ( ( _I |` ( dom R u. ran R ) ) C_ ( t*rec ` R ) /\ R C_ ( t*rec ` R ) /\ ( ( t*rec ` R ) o. ( t*rec ` R ) ) C_ ( t*rec ` R ) ) ) )
36	26, 34, 35	sylancr 695	. . . . . . 7 |- ( ph -> ( ( t*rec ` R ) e. { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } <-> ( ( _I |` ( dom R u. ran R ) ) C_ ( t*rec ` R ) /\ R C_ ( t*rec ` R ) /\ ( ( t*rec ` R ) o. ( t*rec ` R ) ) C_ ( t*rec ` R ) ) ) )
37	23, 24, 25, 36	mpbir3and 1245	. . . . . 6 |- ( ph -> ( t*rec ` R ) e. { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } )
38		ne0i 3921	. . . . . 6 |- ( ( t*rec ` R ) e. { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } -> { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } =/= (/) )
39	37, 38	syl 17	. . . . 5 |- ( ph -> { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } =/= (/) )
40		intex 4820	. . . . 5 |- ( { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } =/= (/) <-> |^| { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } e. _V )
41	39, 40	sylib 208	. . . 4 |- ( ph -> |^| { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } e. _V )
42	1, 12, 13, 41	fvmptd 6288	. . 3 |- ( ph -> ( ( x e. _V |-> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } ) ` R ) = |^| { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } )
43		intss1 4492	. . . . 5 |- ( ( t*rec ` R ) e. { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } -> |^| { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } C_ ( t*rec ` R ) )
44	37, 43	syl 17	. . . 4 |- ( ph -> |^| { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } C_ ( t*rec ` R ) )
45		vex 3203	. . . . . . . 8 |- s e. _V
46		sseq2 3627	. . . . . . . . 9 |- ( z = s -> ( ( _I |` ( dom R u. ran R ) ) C_ z <-> ( _I |` ( dom R u. ran R ) ) C_ s ) )
47		sseq2 3627	. . . . . . . . 9 |- ( z = s -> ( R C_ z <-> R C_ s ) )
48		id 22	. . . . . . . . . . 11 |- ( z = s -> z = s )
49	48, 48	coeq12d 5286	. . . . . . . . . 10 |- ( z = s -> ( z o. z ) = ( s o. s ) )
50	49, 48	sseq12d 3634	. . . . . . . . 9 |- ( z = s -> ( ( z o. z ) C_ z <-> ( s o. s ) C_ s ) )
51	46, 47, 50	3anbi123d 1399	. . . . . . . 8 |- ( z = s -> ( ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) <-> ( ( _I |` ( dom R u. ran R ) ) C_ s /\ R C_ s /\ ( s o. s ) C_ s ) ) )
52	45, 51	elab 3350	. . . . . . 7 |- ( s e. { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } <-> ( ( _I |` ( dom R u. ran R ) ) C_ s /\ R C_ s /\ ( s o. s ) C_ s ) )
53	14, 13	rtrclreclem4 13801	. . . . . . . 8 |- ( ph -> A. s ( ( ( _I |` ( dom R u. ran R ) ) C_ s /\ R C_ s /\ ( s o. s ) C_ s ) -> ( t*rec ` R ) C_ s ) )
54	53	19.21bi 2059	. . . . . . 7 |- ( ph -> ( ( ( _I |` ( dom R u. ran R ) ) C_ s /\ R C_ s /\ ( s o. s ) C_ s ) -> ( t*rec ` R ) C_ s ) )
55	52, 54	syl5bi 232	. . . . . 6 |- ( ph -> ( s e. { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } -> ( t*rec ` R ) C_ s ) )
56	55	ralrimiv 2965	. . . . 5 |- ( ph -> A. s e. { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } ( t*rec ` R ) C_ s )
57		ssint 4493	. . . . 5 |- ( ( t*rec ` R ) C_ |^| { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } <-> A. s e. { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } ( t*rec ` R ) C_ s )
58	56, 57	sylibr 224	. . . 4 |- ( ph -> ( t*rec ` R ) C_ |^| { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } )
59	44, 58	eqssd 3620	. . 3 |- ( ph -> |^| { z | ( ( _I |` ( dom R u. ran R ) ) C_ z /\ R C_ z /\ ( z o. z ) C_ z ) } = ( t*rec ` R ) )
60	42, 59	eqtrd 2656	. 2 |- ( ph -> ( ( x e. _V |-> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } ) ` R ) = ( t*rec ` R ) )
61		df-rtrcl 13727	. . 3 |- t* = ( x e. _V |-> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } )
62		fveq1 6190	. . . . 5 |- ( t* = ( x e. _V |-> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } ) -> ( t* ` R ) = ( ( x e. _V |-> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } ) ` R ) )
63	62	eqeq1d 2624	. . . 4 |- ( t* = ( x e. _V |-> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } ) -> ( ( t* ` R ) = ( t*rec ` R ) <-> ( ( x e. _V |-> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } ) ` R ) = ( t*rec ` R ) ) )
64	63	imbi2d 330	. . 3 |- ( t* = ( x e. _V |-> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } ) -> ( ( ph -> ( t* ` R ) = ( t*rec ` R ) ) <-> ( ph -> ( ( x e. _V |-> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } ) ` R ) = ( t*rec ` R ) ) ) )
65	61, 64	ax-mp 5	. 2 |- ( ( ph -> ( t* ` R ) = ( t*rec ` R ) ) <-> ( ph -> ( ( x e. _V |-> |^| { z | ( ( _I |` ( dom x u. ran x ) ) C_ z /\ x C_ z /\ ( z o. z ) C_ z ) } ) ` R ) = ( t*rec ` R ) ) )
66	60, 65	mpbir 221	1 |- ( ph -> ( t* ` R ) = ( t*rec ` R ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   _Vcvv 3200   u. cun 3572   C_ wss 3574   (/)c0 3915   U.cuni 4436   |^|cint 4475   |-> cmpt 4729   _I cid 5023   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   Rel wrel 5119   ` cfv 5888   t*crtcl 13725   t*reccrtrcl 13795

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-rep 4771  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-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-rtrcl 13727  df-relexp 13761  df-rtrclrec 13796

This theorem is referenced by:  rtrclind  13805