Theorem rngoisoval 30307

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 6304	. . 3 |- ( ( r = R /\ s = S ) -> ( r RngHom s ) = ( R RngHom S ) )
2		fveq2 5872	. . . . . . . 8 |- ( r = R -> ( 1st ` r ) = ( 1st ` R ) )
3		rngisoval.1	. . . . . . . 8 |- G = ( 1st ` R )
4	2, 3	syl6eqr 2526	. . . . . . 7 |- ( r = R -> ( 1st ` r ) = G )
5	4	rneqd 5236	. . . . . 6 |- ( r = R -> ran ( 1st ` r ) = ran G )
6		rngisoval.2	. . . . . 6 |- X = ran G
7	5, 6	syl6eqr 2526	. . . . 5 |- ( r = R -> ran ( 1st ` r ) = X )
8		f1oeq2 5814	. . . . 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 5872	. . . . . . . 8 |- ( s = S -> ( 1st ` s ) = ( 1st ` S ) )
11		rngisoval.3	. . . . . . . 8 |- J = ( 1st ` S )
12	10, 11	syl6eqr 2526	. . . . . . 7 |- ( s = S -> ( 1st ` s ) = J )
13	12	rneqd 5236	. . . . . 6 |- ( s = S -> ran ( 1st ` s ) = ran J )
14		rngisoval.4	. . . . . 6 |- Y = ran J
15	13, 14	syl6eqr 2526	. . . . 5 |- ( s = S -> ran ( 1st ` s ) = Y )
16		f1oeq3 5815	. . . . 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 3113	. 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 30306	. 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 6320	. . 3 |- ( R RngHom S ) e. _V
22	21	rabex 4604	. 2 |- { f e. ( R RngHom S ) | f : X -1-1-onto-> Y } e. _V
23	19, 20, 22	ovmpt2a 6428	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 1379   e. wcel 1767   {crab 2821   ran crn 5006   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   1stc1st 6793   RingOpscrngo 25200   RngHom crnghom 30290   RngIso crngiso 30291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-rngoiso 30306

This theorem is referenced by:  isrngoiso  30308