Theorem efgi2 18138

Description: Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)

Hypotheses

Ref	Expression
efgval.w	|- W = ( _I ` Word ( I X. 2o ) )
efgval.r	|- .~ = ( ~FG ` I )
efgval2.m	|- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
efgval2.t	|- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )

Assertion

Ref	Expression
efgi2	|- ( ( A e. W /\ B e. ran ( T ` A ) ) -> A .~ B )

Distinct variable groups:   y, z   v, n, w, y, z   n, M, v, w   n, W, v, w, y, z   y, .~ , z   n, I, v, w, y, z

Allowed substitution hints:   A( y, z, w, v, n)   B( y, z, w, v, n)   .~ ( w, v, n)   T( y, z, w, v, n)   M( y, z)

Proof of Theorem efgi2

Dummy variables a r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . . . . . 11 |- ( a = A -> ( T ` a ) = ( T ` A ) )
2	1	rneqd 5353	. . . . . . . . . 10 |- ( a = A -> ran ( T ` a ) = ran ( T ` A ) )
3		eceq1 7782	. . . . . . . . . 10 |- ( a = A -> [ a ] r = [ A ] r )
4	2, 3	sseq12d 3634	. . . . . . . . 9 |- ( a = A -> ( ran ( T ` a ) C_ [ a ] r <-> ran ( T ` A ) C_ [ A ] r ) )
5	4	rspcv 3305	. . . . . . . 8 |- ( A e. W -> ( A. a e. W ran ( T ` a ) C_ [ a ] r -> ran ( T ` A ) C_ [ A ] r ) )
6	5	adantr 481	. . . . . . 7 |- ( ( A e. W /\ B e. ran ( T ` A ) ) -> ( A. a e. W ran ( T ` a ) C_ [ a ] r -> ran ( T ` A ) C_ [ A ] r ) )
7		ssel 3597	. . . . . . . . 9 |- ( ran ( T ` A ) C_ [ A ] r -> ( B e. ran ( T ` A ) -> B e. [ A ] r ) )
8	7	com12 32	. . . . . . . 8 |- ( B e. ran ( T ` A ) -> ( ran ( T ` A ) C_ [ A ] r -> B e. [ A ] r ) )
9		simpl 473	. . . . . . . . . . 11 |- ( ( B e. [ A ] r /\ A e. W ) -> B e. [ A ] r )
10		elecg 7785	. . . . . . . . . . 11 |- ( ( B e. [ A ] r /\ A e. W ) -> ( B e. [ A ] r <-> A r B ) )
11	9, 10	mpbid 222	. . . . . . . . . 10 |- ( ( B e. [ A ] r /\ A e. W ) -> A r B )
12		df-br 4654	. . . . . . . . . 10 |- ( A r B <-> <. A , B >. e. r )
13	11, 12	sylib 208	. . . . . . . . 9 |- ( ( B e. [ A ] r /\ A e. W ) -> <. A , B >. e. r )
14	13	expcom 451	. . . . . . . 8 |- ( A e. W -> ( B e. [ A ] r -> <. A , B >. e. r ) )
15	8, 14	sylan9r 690	. . . . . . 7 |- ( ( A e. W /\ B e. ran ( T ` A ) ) -> ( ran ( T ` A ) C_ [ A ] r -> <. A , B >. e. r ) )
16	6, 15	syld 47	. . . . . 6 |- ( ( A e. W /\ B e. ran ( T ` A ) ) -> ( A. a e. W ran ( T ` a ) C_ [ a ] r -> <. A , B >. e. r ) )
17	16	adantld 483	. . . . 5 |- ( ( A e. W /\ B e. ran ( T ` A ) ) -> ( ( r Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] r ) -> <. A , B >. e. r ) )
18	17	alrimiv 1855	. . . 4 |- ( ( A e. W /\ B e. ran ( T ` A ) ) -> A. r ( ( r Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] r ) -> <. A , B >. e. r ) )
19		opex 4932	. . . . 5 |- <. A , B >. e. _V
20	19	elintab 4487	. . . 4 |- ( <. A , B >. e. |^| { r | ( r Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] r ) } <-> A. r ( ( r Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] r ) -> <. A , B >. e. r ) )
21	18, 20	sylibr 224	. . 3 |- ( ( A e. W /\ B e. ran ( T ` A ) ) -> <. A , B >. e. |^| { r | ( r Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] r ) } )
22		efgval.w	. . . 4 |- W = ( _I ` Word ( I X. 2o ) )
23		efgval.r	. . . 4 |- .~ = ( ~FG ` I )
24		efgval2.m	. . . 4 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
25		efgval2.t	. . . 4 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
26	22, 23, 24, 25	efgval2 18137	. . 3 |- .~ = |^| { r | ( r Er W /\ A. a e. W ran ( T ` a ) C_ [ a ] r ) }
27	21, 26	syl6eleqr 2712	. 2 |- ( ( A e. W /\ B e. ran ( T ` A ) ) -> <. A , B >. e. .~ )
28		df-br 4654	. 2 |- ( A .~ B <-> <. A , B >. e. .~ )
29	27, 28	sylibr 224	1 |- ( ( A e. W /\ B e. ran ( T ` A ) ) -> A .~ B )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   \ cdif 3571   C_ wss 3574   <.cop 4183   <.cotp 4185   |^|cint 4475   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   Er wer 7739   [cec 7740   0cc0 9936   ...cfz 12326   #chash 13117  Word cword 13291   splice csplice 13296   <"cs2 13586   ~_FG cefg 18119

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-ot 4186  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-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-s2 13593  df-efg 18122

This theorem is referenced by:  efginvrel2  18140  efgsrel  18147  efgcpbllemb  18168