Theorem rngoisoval 28806

Description: The set of ring isomorphisms. (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
rngoisoval	|- ( ( R e. RingOps /\ S e. RingOps ) -> ( R RngIso S ) = { f e. ( R RngHom S ) | f : X -1-1-onto-> Y } )

Distinct variable groups:   R, f   S, f   f, X   f, Y

Allowed substitution hints:   G( f)   J( f)

Proof of Theorem rngoisoval

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

Step	Hyp	Ref	Expression
1		oveq12 6119	. . 3 |- ( ( r = R /\ s = S ) -> ( r RngHom s ) = ( R RngHom S ) )
2		fveq2 5710	. . . . . . . 8 |- ( r = R -> ( 1st ` r ) = ( 1st ` R ) )
3		rngisoval.1	. . . . . . . 8 |- G = ( 1st ` R )
4	2, 3	syl6eqr 2493	. . . . . . 7 |- ( r = R -> ( 1st ` r ) = G )
5	4	rneqd 5086	. . . . . 6 |- ( r = R -> ran ( 1st ` r ) = ran G )
6		rngisoval.2	. . . . . 6 |- X = ran G
7	5, 6	syl6eqr 2493	. . . . 5 |- ( r = R -> ran ( 1st ` r ) = X )
8		f1oeq2 5652	. . . . 5 |- ( ran ( 1st ` r ) = X -> ( f : ran ( 1st ` r ) -1-1-onto-> ran ( 1st ` s ) <-> f : X -1-1-onto-> ran ( 1st ` s ) ) )
9	7, 8	syl 16	. . . 4 |- ( r = R -> ( f : ran ( 1st ` r ) -1-1-onto-> ran ( 1st ` s ) <-> f : X -1-1-onto-> ran ( 1st ` s ) ) )
10		fveq2 5710	. . . . . . . 8 |- ( s = S -> ( 1st ` s ) = ( 1st ` S ) )
11		rngisoval.3	. . . . . . . 8 |- J = ( 1st ` S )
12	10, 11	syl6eqr 2493	. . . . . . 7 |- ( s = S -> ( 1st ` s ) = J )
13	12	rneqd 5086	. . . . . 6 |- ( s = S -> ran ( 1st ` s ) = ran J )
14		rngisoval.4	. . . . . 6 |- Y = ran J
15	13, 14	syl6eqr 2493	. . . . 5 |- ( s = S -> ran ( 1st ` s ) = Y )
16		f1oeq3 5653	. . . . 5 |- ( ran ( 1st ` s ) = Y -> ( f : X -1-1-onto-> ran ( 1st ` s ) <-> f : X -1-1-onto-> Y ) )
17	15, 16	syl 16	. . . 4 |- ( s = S -> ( f : X -1-1-onto-> ran ( 1st ` s ) <-> f : X -1-1-onto-> Y ) )
18	9, 17	sylan9bb 699	. . 3 |- ( ( r = R /\ s = S ) -> ( f : ran ( 1st ` r ) -1-1-onto-> ran ( 1st ` s ) <-> f : X -1-1-onto-> Y ) )
19	1, 18	rabeqbidv 2986	. 2 |- ( ( r = R /\ s = S ) -> { f e. ( r RngHom s ) | f : ran ( 1st ` r ) -1-1-onto-> ran ( 1st ` s ) } = { f e. ( R RngHom S ) | f : X -1-1-onto-> Y } )
20		df-rngoiso 28805	. 2 |- RngIso = ( r e. RingOps , s e. RingOps |-> { f e. ( r RngHom s ) | f : ran ( 1st ` r ) -1-1-onto-> ran ( 1st ` s ) } )
21		ovex 6135	. . 3 |- ( R RngHom S ) e. _V
22	21	rabex 4462	. 2 |- { f e. ( R RngHom S ) | f : X -1-1-onto-> Y } e. _V
23	19, 20, 22	ovmpt2a 6240	1 |- ( ( R e. RingOps /\ S e. RingOps ) -> ( R RngIso S ) = { f e. ( R RngHom S ) | f : X -1-1-onto-> Y } )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   {crab 2738   ran crn 4860   -1-1-onto->wf1o 5436   ` cfv 5437  (class class class)co 6110   1stc1st 6594   RingOpscrngo 23881   RngHom crnghom 28789   RngIso crngiso 28790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-br 4312  df-opab 4370  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-rngoiso 28805

This theorem is referenced by:  isrngoiso  28807