Theorem rngoisoval 32260

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 6323	. . 3 |- ( ( r = R /\ s = S ) -> ( r RngHom s ) = ( R RngHom S ) )
2		fveq2 5887	. . . . . . . 8 |- ( r = R -> ( 1st ` r ) = ( 1st ` R ) )
3		rngisoval.1	. . . . . . . 8 |- G = ( 1st ` R )
4	2, 3	syl6eqr 2513	. . . . . . 7 |- ( r = R -> ( 1st ` r ) = G )
5	4	rneqd 5080	. . . . . 6 |- ( r = R -> ran ( 1st ` r ) = ran G )
6		rngisoval.2	. . . . . 6 |- X = ran G
7	5, 6	syl6eqr 2513	. . . . 5 |- ( r = R -> ran ( 1st ` r ) = X )
8		f1oeq2 5828	. . . . 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 5887	. . . . . . . 8 |- ( s = S -> ( 1st ` s ) = ( 1st ` S ) )
11		rngisoval.3	. . . . . . . 8 |- J = ( 1st ` S )
12	10, 11	syl6eqr 2513	. . . . . . 7 |- ( s = S -> ( 1st ` s ) = J )
13	12	rneqd 5080	. . . . . 6 |- ( s = S -> ran ( 1st ` s ) = ran J )
14		rngisoval.4	. . . . . 6 |- Y = ran J
15	13, 14	syl6eqr 2513	. . . . 5 |- ( s = S -> ran ( 1st ` s ) = Y )
16		f1oeq3 5829	. . . . 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 711	. . 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 3051	. 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 32259	. 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 6342	. . 3 |- ( R RngHom S ) e. _V
22	21	rabex 4567	. 2 |- { f e. ( R RngHom S ) | f : X -1-1-onto-> Y } e. _V
23	19, 20, 22	ovmpt2a 6453	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 189   /\ wa 375   = wceq 1454   e. wcel 1897   {crab 2752   ran crn 4853   -1-1-onto->wf1o 5599   ` cfv 5600  (class class class)co 6314   1stc1st 6817   RingOpscrngo 26151   RngHom crnghom 32243   RngIso crngiso 32244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-rngoiso 32259

This theorem is referenced by:  isrngoiso  32261