Theorem rngokerinj 33774

Description: A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)

Hypotheses

Ref	Expression
rngkerinj.1	|- G = ( 1st ` R )
rngkerinj.2	|- X = ran G
rngkerinj.3	|- W = (GId ` G )
rngkerinj.4	|- J = ( 1st ` S )
rngkerinj.5	|- Y = ran J
rngkerinj.6	|- Z = (GId ` J )

Assertion

Ref	Expression
rngokerinj	|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( F : X -1-1-> Y <-> ( `' F " { Z } ) = { W } ) )

Proof of Theorem rngokerinj

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( 1st ` R ) = ( 1st ` R )
2	1	rngogrpo 33709	. . 3 |- ( R e. RingOps -> ( 1st ` R ) e. GrpOp )
3	2	3ad2ant1 1082	. 2 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( 1st ` R ) e. GrpOp )
4		eqid 2622	. . . 4 |- ( 1st ` S ) = ( 1st ` S )
5	4	rngogrpo 33709	. . 3 |- ( S e. RingOps -> ( 1st ` S ) e. GrpOp )
6	5	3ad2ant2 1083	. 2 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( 1st ` S ) e. GrpOp )
7	1, 4	rngogrphom 33770	. 2 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> F e. ( ( 1st ` R ) GrpOpHom ( 1st ` S ) ) )
8		rngkerinj.2	. . . 4 |- X = ran G
9		rngkerinj.1	. . . . 5 |- G = ( 1st ` R )
10	9	rneqi 5352	. . . 4 |- ran G = ran ( 1st ` R )
11	8, 10	eqtri 2644	. . 3 |- X = ran ( 1st ` R )
12		rngkerinj.3	. . . 4 |- W = (GId ` G )
13	9	fveq2i 6194	. . . 4 |- (GId ` G ) = (GId ` ( 1st ` R ) )
14	12, 13	eqtri 2644	. . 3 |- W = (GId ` ( 1st ` R ) )
15		rngkerinj.5	. . . 4 |- Y = ran J
16		rngkerinj.4	. . . . 5 |- J = ( 1st ` S )
17	16	rneqi 5352	. . . 4 |- ran J = ran ( 1st ` S )
18	15, 17	eqtri 2644	. . 3 |- Y = ran ( 1st ` S )
19		rngkerinj.6	. . . 4 |- Z = (GId ` J )
20	16	fveq2i 6194	. . . 4 |- (GId ` J ) = (GId ` ( 1st ` S ) )
21	19, 20	eqtri 2644	. . 3 |- Z = (GId ` ( 1st ` S ) )
22	11, 14, 18, 21	grpokerinj 33692	. 2 |- ( ( ( 1st ` R ) e. GrpOp /\ ( 1st ` S ) e. GrpOp /\ F e. ( ( 1st ` R ) GrpOpHom ( 1st ` S ) ) ) -> ( F : X -1-1-> Y <-> ( `' F " { Z } ) = { W } ) )
23	3, 6, 7, 22	syl3anc 1326	1 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> ( F : X -1-1-> Y <-> ( `' F " { Z } ) = { W } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   {csn 4177   `'ccnv 5113   ran crn 5115   "cima 5117   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   1stc1st 7166   GrpOpcgr 27343  GIdcgi 27344   GrpOpHom cghomOLD 33682   RingOpscrngo 33693   RngHom crnghom 33759

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

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-pw 4160  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-ghomOLD 33683  df-rngo 33694  df-rngohom 33762

This theorem is referenced by: (None)