Theorem rngoisoval 33776

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 6659	. . 3 |- ( ( r = R /\ s = S ) -> ( r RngHom s ) = ( R RngHom S ) )
2		fveq2 6191	. . . . . . . 8 |- ( r = R -> ( 1st ` r ) = ( 1st ` R ) )
3		rngisoval.1	. . . . . . . 8 |- G = ( 1st ` R )
4	2, 3	syl6eqr 2674	. . . . . . 7 |- ( r = R -> ( 1st ` r ) = G )
5	4	rneqd 5353	. . . . . 6 |- ( r = R -> ran ( 1st ` r ) = ran G )
6		rngisoval.2	. . . . . 6 |- X = ran G
7	5, 6	syl6eqr 2674	. . . . 5 |- ( r = R -> ran ( 1st ` r ) = X )
8		f1oeq2 6128	. . . . 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 17	. . . 4 |- ( r = R -> ( f : ran ( 1st ` r ) -1-1-onto-> ran ( 1st ` s ) <-> f : X -1-1-onto-> ran ( 1st ` s ) ) )
10		fveq2 6191	. . . . . . . 8 |- ( s = S -> ( 1st ` s ) = ( 1st ` S ) )
11		rngisoval.3	. . . . . . . 8 |- J = ( 1st ` S )
12	10, 11	syl6eqr 2674	. . . . . . 7 |- ( s = S -> ( 1st ` s ) = J )
13	12	rneqd 5353	. . . . . 6 |- ( s = S -> ran ( 1st ` s ) = ran J )
14		rngisoval.4	. . . . . 6 |- Y = ran J
15	13, 14	syl6eqr 2674	. . . . 5 |- ( s = S -> ran ( 1st ` s ) = Y )
16		f1oeq3 6129	. . . . 5 |- ( ran ( 1st ` s ) = Y -> ( f : X -1-1-onto-> ran ( 1st ` s ) <-> f : X -1-1-onto-> Y ) )
17	15, 16	syl 17	. . . 4 |- ( s = S -> ( f : X -1-1-onto-> ran ( 1st ` s ) <-> f : X -1-1-onto-> Y ) )
18	9, 17	sylan9bb 736	. . 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 3195	. 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 33775	. 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 6678	. . 3 |- ( R RngHom S ) e. _V
22	21	rabex 4813	. 2 |- { f e. ( R RngHom S ) | f : X -1-1-onto-> Y } e. _V
23	19, 20, 22	ovmpt2a 6791	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 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:  isrngoiso  33777