Theorem rngohomf 33765

Description: A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.)

Hypotheses

Ref	Expression
rnghomf.1	|- G = ( 1st ` R )
rnghomf.2	|- X = ran G
rnghomf.3	|- J = ( 1st ` S )
rnghomf.4	|- Y = ran J

Assertion

Ref	Expression
rngohomf	|- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> F : X --> Y )

Proof of Theorem rngohomf

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

Step	Hyp	Ref	Expression
1		rnghomf.1	. . . . 5 |- G = ( 1st ` R )
2		eqid 2622	. . . . 5 |- ( 2nd ` R ) = ( 2nd ` R )
3		rnghomf.2	. . . . 5 |- X = ran G
4		eqid 2622	. . . . 5 |- (GId ` ( 2nd ` R ) ) = (GId ` ( 2nd ` R ) )
5		rnghomf.3	. . . . 5 |- J = ( 1st ` S )
6		eqid 2622	. . . . 5 |- ( 2nd ` S ) = ( 2nd ` S )
7		rnghomf.4	. . . . 5 |- Y = ran J
8		eqid 2622	. . . . 5 |- (GId ` ( 2nd ` S ) ) = (GId ` ( 2nd ` S ) )
9	1, 2, 3, 4, 5, 6, 7, 8	isrngohom 33764	. . . 4 |- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RngHom S ) <-> ( F : X --> Y /\ ( F ` (GId ` ( 2nd ` R ) ) ) = (GId ` ( 2nd ` S ) ) /\ A. x e. X A. y e. X ( ( F ` ( x G y ) ) = ( ( F ` x ) J ( F ` y ) ) /\ ( F ` ( x ( 2nd ` R ) y ) ) = ( ( F ` x ) ( 2nd ` S ) ( F ` y ) ) ) ) ) )
10	9	biimpa 501	. . 3 |- ( ( ( R e. RingOps /\ S e. RingOps ) /\ F e. ( R RngHom S ) ) -> ( F : X --> Y /\ ( F ` (GId ` ( 2nd ` R ) ) ) = (GId ` ( 2nd ` S ) ) /\ A. x e. X A. y e. X ( ( F ` ( x G y ) ) = ( ( F ` x ) J ( F ` y ) ) /\ ( F ` ( x ( 2nd ` R ) y ) ) = ( ( F ` x ) ( 2nd ` S ) ( F ` y ) ) ) ) )
11	10	simp1d 1073	. 2 |- ( ( ( R e. RingOps /\ S e. RingOps ) /\ F e. ( R RngHom S ) ) -> F : X --> Y )
12	11	3impa 1259	1 |- ( ( R e. RingOps /\ S e. RingOps /\ F e. ( R RngHom S ) ) -> F : X --> Y )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167  GIdcgi 27344   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-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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-rngohom 33762

This theorem is referenced by:  rngohomcl  33766  rngogrphom  33770  rngohomco  33773  keridl  33831