Theorem isrngoiso 33777

Description: The predicate "is a ring isomorphism between R and S." (Contributed by Jeff Madsen, 16-Jun-2011.)

Hypotheses

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

Assertion

Ref	Expression
isrngoiso	|- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RngIso S ) <-> ( F e. ( R RngHom S ) /\ F : X -1-1-onto-> Y ) ) )

Proof of Theorem isrngoiso

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngisoval.1	. . . 4 |- G = ( 1st ` R )
2		rngisoval.2	. . . 4 |- X = ran G
3		rngisoval.3	. . . 4 |- J = ( 1st ` S )
4		rngisoval.4	. . . 4 |- Y = ran J
5	1, 2, 3, 4	rngoisoval 33776	. . 3 |- ( ( R e. RingOps /\ S e. RingOps ) -> ( R RngIso S ) = { f e. ( R RngHom S ) | f : X -1-1-onto-> Y } )
6	5	eleq2d 2687	. 2 |- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RngIso S ) <-> F e. { f e. ( R RngHom S ) | f : X -1-1-onto-> Y } ) )
7		f1oeq1 6127	. . 3 |- ( f = F -> ( f : X -1-1-onto-> Y <-> F : X -1-1-onto-> Y ) )
8	7	elrab 3363	. 2 |- ( F e. { f e. ( R RngHom S ) | f : X -1-1-onto-> Y } <-> ( F e. ( R RngHom S ) /\ F : X -1-1-onto-> Y ) )
9	6, 8	syl6bb 276	1 |- ( ( R e. RingOps /\ S e. RingOps ) -> ( F e. ( R RngIso S ) <-> ( F e. ( R RngHom S ) /\ F : X -1-1-onto-> Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   ran crn 5115   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   1stc1st 7166   RingOpscrngo 33693   RngHom crnghom 33759   RngIso crngiso 33760

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-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-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-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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rngoiso 33775

This theorem is referenced by:  rngoiso1o  33778  rngoisohom  33779  rngoisocnv  33780  rngoisoco  33781