Theorem vrgpfval 18179

Description: The canonical injection from the generating set I to the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
vrgpfval.r	|- .~ = ( ~FG ` I )
vrgpfval.u	|- U = (varFGrp ` I )

Assertion

Ref	Expression
vrgpfval	|- ( I e. V -> U = ( j e. I |-> [ <" <. j , (/) >. "> ] .~ ) )

Distinct variable groups:   j, I   .~ , j   j, V

Allowed substitution hint:   U( j)

Proof of Theorem vrgpfval

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vrgpfval.u	. 2 |- U = (varFGrp ` I )
2		elex 3212	. . 3 |- ( I e. V -> I e. _V )
3		id 22	. . . . 5 |- ( i = I -> i = I )
4		fveq2 6191	. . . . . . 7 |- ( i = I -> ( ~FG ` i ) = ( ~FG ` I ) )
5		vrgpfval.r	. . . . . . 7 |- .~ = ( ~FG ` I )
6	4, 5	syl6eqr 2674	. . . . . 6 |- ( i = I -> ( ~FG ` i ) = .~ )
7		eceq2 7784	. . . . . 6 |- ( ( ~FG ` i ) = .~ -> [ <" <. j , (/) >. "> ] ( ~FG ` i ) = [ <" <. j , (/) >. "> ] .~ )
8	6, 7	syl 17	. . . . 5 |- ( i = I -> [ <" <. j , (/) >. "> ] ( ~FG ` i ) = [ <" <. j , (/) >. "> ] .~ )
9	3, 8	mpteq12dv 4733	. . . 4 |- ( i = I -> ( j e. i |-> [ <" <. j , (/) >. "> ] ( ~FG ` i ) ) = ( j e. I |-> [ <" <. j , (/) >. "> ] .~ ) )
10		df-vrgp 18124	. . . 4 |- varFGrp = ( i e. _V |-> ( j e. i |-> [ <" <. j , (/) >. "> ] ( ~FG ` i ) ) )
11		vex 3203	. . . . 5 |- i e. _V
12	11	mptex 6486	. . . 4 |- ( j e. i |-> [ <" <. j , (/) >. "> ] ( ~FG ` i ) ) e. _V
13	9, 10, 12	fvmpt3i 6287	. . 3 |- ( I e. _V -> (varFGrp ` I ) = ( j e. I |-> [ <" <. j , (/) >. "> ] .~ ) )
14	2, 13	syl 17	. 2 |- ( I e. V -> (varFGrp ` I ) = ( j e. I |-> [ <" <. j , (/) >. "> ] .~ ) )
15	1, 14	syl5eq 2668	1 |- ( I e. V -> U = ( j e. I |-> [ <" <. j , (/) >. "> ] .~ ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   <.cop 4183   |-> cmpt 4729   ` cfv 5888   [cec 7740   <"cs1 13294   ~_FG cefg 18119  var_FGrpcvrgp 18121

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-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-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ec 7744  df-vrgp 18124

This theorem is referenced by:  vrgpval  18180  vrgpf  18181