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Theorem rngoisocnv 28955
Description: The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
rngoisocnv  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  `' F  e.  ( S  RngIso  R ) )

Proof of Theorem rngoisocnv
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1ocnv 5764 . . . . . . . 8  |-  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  `' F : ran  ( 1st `  S
)
-1-1-onto-> ran  ( 1st `  R
) )
2 f1of 5752 . . . . . . . 8  |-  ( `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R )  ->  `' F : ran  ( 1st `  S ) --> ran  ( 1st `  R
) )
31, 2syl 16 . . . . . . 7  |-  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  `' F : ran  ( 1st `  S
) --> ran  ( 1st `  R ) )
43ad2antll 728 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  ->  `' F : ran  ( 1st `  S ) --> ran  ( 1st `  R
) )
5 eqid 2454 . . . . . . . . . 10  |-  ( 2nd `  R )  =  ( 2nd `  R )
6 eqid 2454 . . . . . . . . . 10  |-  (GId `  ( 2nd `  R ) )  =  (GId `  ( 2nd `  R ) )
7 eqid 2454 . . . . . . . . . 10  |-  ( 2nd `  S )  =  ( 2nd `  S )
8 eqid 2454 . . . . . . . . . 10  |-  (GId `  ( 2nd `  S ) )  =  (GId `  ( 2nd `  S ) )
95, 6, 7, 8rngohom1 28942 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) ) )
1093expa 1188 . . . . . . . 8  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) ) )
1110adantrr 716 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) ) )
12 eqid 2454 . . . . . . . . . . 11  |-  ran  ( 1st `  R )  =  ran  ( 1st `  R
)
1312, 5, 6rngo1cl 24088 . . . . . . . . . 10  |-  ( R  e.  RingOps  ->  (GId `  ( 2nd `  R ) )  e.  ran  ( 1st `  R ) )
14 f1ocnvfv 6097 . . . . . . . . . 10  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  (GId `  ( 2nd `  R
) )  e.  ran  ( 1st `  R ) )  ->  ( ( F `  (GId `  ( 2nd `  R ) ) )  =  (GId `  ( 2nd `  S ) )  ->  ( `' F `  (GId `  ( 2nd `  S ) ) )  =  (GId `  ( 2nd `  R ) ) ) )
1513, 14sylan2 474 . . . . . . . . 9  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  R  e.  RingOps )  -> 
( ( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) )  -> 
( `' F `  (GId `  ( 2nd `  S
) ) )  =  (GId `  ( 2nd `  R ) ) ) )
1615ancoms 453 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  (
( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) )  -> 
( `' F `  (GId `  ( 2nd `  S
) ) )  =  (GId `  ( 2nd `  R ) ) ) )
1716ad2ant2rl 748 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( ( F `  (GId `  ( 2nd `  R
) ) )  =  (GId `  ( 2nd `  S ) )  -> 
( `' F `  (GId `  ( 2nd `  S
) ) )  =  (GId `  ( 2nd `  R ) ) ) )
1811, 17mpd 15 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( `' F `  (GId `  ( 2nd `  S
) ) )  =  (GId `  ( 2nd `  R ) ) )
19 f1ocnvfv2 6096 . . . . . . . . . . . . . 14  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  x  e.  ran  ( 1st `  S ) )  -> 
( F `  ( `' F `  x ) )  =  x )
20 f1ocnvfv2 6096 . . . . . . . . . . . . . 14  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  ( `' F `  y ) )  =  y )
2119, 20anim12da 28772 . . . . . . . . . . . . 13  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) )  =  x  /\  ( F `  ( `' F `  y ) )  =  y ) )
22 oveq12 6212 . . . . . . . . . . . . 13  |-  ( ( ( F `  ( `' F `  x ) )  =  x  /\  ( F `  ( `' F `  y ) )  =  y )  ->  ( ( F `
 ( `' F `  x ) ) ( 1st `  S ) ( F `  ( `' F `  y ) ) )  =  ( x ( 1st `  S
) y ) )
2321, 22syl 16 . . . . . . . . . . . 12  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) )  =  ( x ( 1st `  S
) y ) )
2423adantll 713 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( F `
 ( `' F `  x ) ) ( 1st `  S ) ( F `  ( `' F `  y ) ) )  =  ( x ( 1st `  S
) y ) )
2524adantll 713 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) )  =  ( x ( 1st `  S
) y ) )
26 f1ocnvdm 6101 . . . . . . . . . . . . . . . 16  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  x  e.  ran  ( 1st `  S ) )  -> 
( `' F `  x )  e.  ran  ( 1st `  R ) )
27 f1ocnvdm 6101 . . . . . . . . . . . . . . . 16  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( `' F `  y )  e.  ran  ( 1st `  R ) )
2826, 27anim12da 28772 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )
29 eqid 2454 . . . . . . . . . . . . . . . 16  |-  ( 1st `  R )  =  ( 1st `  R )
30 eqid 2454 . . . . . . . . . . . . . . . 16  |-  ( 1st `  S )  =  ( 1st `  S )
3129, 12, 30rngohomadd 28943 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( ( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )  ->  ( F `  ( ( `' F `  x ) ( 1st `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) ) )
3228, 31sylan2 474 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  /\  ( x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) ) ) )  ->  ( F `  ( ( `' F `  x ) ( 1st `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) ) )
3332exp32 605 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
)  ->  ( (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  (
( `' F `  x ) ( 1st `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) ) ) ) )
34333expa 1188 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  ( (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  (
( `' F `  x ) ( 1st `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 1st `  S
) ( F `  ( `' F `  y ) ) ) ) ) )
3534impr 619 . . . . . . . . . . 11  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( ( x  e. 
ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) )  ->  ( F `  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  =  ( ( F `
 ( `' F `  x ) ) ( 1st `  S ) ( F `  ( `' F `  y ) ) ) ) )
3635imp 429 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  =  ( ( F `
 ( `' F `  x ) ) ( 1st `  S ) ( F `  ( `' F `  y ) ) ) )
37 eqid 2454 . . . . . . . . . . . . . . . 16  |-  ran  ( 1st `  S )  =  ran  ( 1st `  S
)
3830, 37rngogcl 24050 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  RingOps  /\  x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) )  ->  (
x ( 1st `  S
) y )  e. 
ran  ( 1st `  S
) )
39383expb 1189 . . . . . . . . . . . . . 14  |-  ( ( S  e.  RingOps  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S ) )
40 f1ocnvfv2 6096 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S ) )  -> 
( F `  ( `' F `  ( x ( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4140ancoms 453 . . . . . . . . . . . . . 14  |-  ( ( ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4239, 41sylan 471 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  RingOps  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4342an32s 802 . . . . . . . . . . . 12  |-  ( ( ( S  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x ( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4443adantlll 717 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x ( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4544adantlrl 719 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( x ( 1st `  S
) y ) )
4625, 36, 453eqtr4rd 2506 . . . . . . . . 9  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) ) )
47 f1of1 5751 . . . . . . . . . . . 12  |-  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  F : ran  ( 1st `  R
) -1-1-> ran  ( 1st `  S
) )
4847ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  F : ran  ( 1st `  R )
-1-1-> ran  ( 1st `  S
) )
49 f1ocnvdm 6101 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S ) )  -> 
( `' F `  ( x ( 1st `  S ) y ) )  e.  ran  ( 1st `  R ) )
5049ancoms 453 . . . . . . . . . . . . . 14  |-  ( ( ( x ( 1st `  S ) y )  e.  ran  ( 1st `  S )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F `  ( x ( 1st `  S
) y ) )  e.  ran  ( 1st `  R ) )
5139, 50sylan 471 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  RingOps  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F `  ( x ( 1st `  S
) y ) )  e.  ran  ( 1st `  R ) )
5251an32s 802 . . . . . . . . . . . 12  |-  ( ( ( S  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 1st `  S ) y ) )  e.  ran  ( 1st `  R ) )
5352adantlll 717 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 1st `  S ) y ) )  e.  ran  ( 1st `  R ) )
5429, 12rngogcl 24050 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  ( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) )  -> 
( ( `' F `  x ) ( 1st `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) )
55543expb 1189 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  (
( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )  ->  (
( `' F `  x ) ( 1st `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) )
5628, 55sylan2 474 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  /\  ( x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) ) ) )  ->  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
5756anassrs 648 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
5857adantllr 718 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
59 f1fveq 6087 . . . . . . . . . . 11  |-  ( ( F : ran  ( 1st `  R ) -1-1-> ran  ( 1st `  S )  /\  ( ( `' F `  ( x ( 1st `  S
) y ) )  e.  ran  ( 1st `  R )  /\  (
( `' F `  x ) ( 1st `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) ) )  ->  ( ( F `  ( `' F `  ( x
( 1st `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) ) )
6048, 53, 58, 59syl12anc 1217 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( F `
 ( `' F `  ( x ( 1st `  S ) y ) ) )  =  ( F `  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) ) )
6160adantlrl 719 . . . . . . . . 9  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  ( x ( 1st `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) ) )
6246, 61mpbid 210 . . . . . . . 8  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 1st `  S
) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) ) )
63 oveq12 6212 . . . . . . . . . . . . 13  |-  ( ( ( F `  ( `' F `  x ) )  =  x  /\  ( F `  ( `' F `  y ) )  =  y )  ->  ( ( F `
 ( `' F `  x ) ) ( 2nd `  S ) ( F `  ( `' F `  y ) ) )  =  ( x ( 2nd `  S
) y ) )
6421, 63syl 16 . . . . . . . . . . . 12  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) )  =  ( x ( 2nd `  S
) y ) )
6564adantll 713 . . . . . . . . . . 11  |-  ( ( ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( F `
 ( `' F `  x ) ) ( 2nd `  S ) ( F `  ( `' F `  y ) ) )  =  ( x ( 2nd `  S
) y ) )
6665adantll 713 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) )  =  ( x ( 2nd `  S
) y ) )
6729, 12, 5, 7rngohommul 28944 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( ( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )  ->  ( F `  ( ( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) ) )
6828, 67sylan2 474 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  /\  ( x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) ) ) )  ->  ( F `  ( ( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) ) )
6968exp32 605 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
)  ->  ( (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  (
( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) ) ) ) )
70693expa 1188 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  F  e.  ( R  RngHom  S ) )  -> 
( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  ->  ( (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) )  -> 
( F `  (
( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) ) )  =  ( ( F `  ( `' F `  x ) ) ( 2nd `  S
) ( F `  ( `' F `  y ) ) ) ) ) )
7170impr 619 . . . . . . . . . . 11  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( ( x  e. 
ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) )  ->  ( F `  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  =  ( ( F `
 ( `' F `  x ) ) ( 2nd `  S ) ( F `  ( `' F `  y ) ) ) ) )
7271imp 429 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  =  ( ( F `
 ( `' F `  x ) ) ( 2nd `  S ) ( F `  ( `' F `  y ) ) ) )
7330, 7, 37rngocl 24041 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  RingOps  /\  x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) )  ->  (
x ( 2nd `  S
) y )  e. 
ran  ( 1st `  S
) )
74733expb 1189 . . . . . . . . . . . . . 14  |-  ( ( S  e.  RingOps  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S ) )
75 f1ocnvfv2 6096 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S ) )  -> 
( F `  ( `' F `  ( x ( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
7675ancoms 453 . . . . . . . . . . . . . 14  |-  ( ( ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
7774, 76sylan 471 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  RingOps  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
7877an32s 802 . . . . . . . . . . . 12  |-  ( ( ( S  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x ( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
7978adantlll 717 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x ( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
8079adantlrl 719 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( x ( 2nd `  S
) y ) )
8166, 72, 803eqtr4rd 2506 . . . . . . . . 9  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
82 f1ocnvdm 6101 . . . . . . . . . . . . . . 15  |-  ( ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S )  /\  ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S ) )  -> 
( `' F `  ( x ( 2nd `  S ) y ) )  e.  ran  ( 1st `  R ) )
8382ancoms 453 . . . . . . . . . . . . . 14  |-  ( ( ( x ( 2nd `  S ) y )  e.  ran  ( 1st `  S )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F `  ( x ( 2nd `  S
) y ) )  e.  ran  ( 1st `  R ) )
8474, 83sylan 471 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  RingOps  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  /\  F : ran  ( 1st `  R
)
-1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F `  ( x ( 2nd `  S
) y ) )  e.  ran  ( 1st `  R ) )
8584an32s 802 . . . . . . . . . . . 12  |-  ( ( ( S  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 2nd `  S ) y ) )  e.  ran  ( 1st `  R ) )
8685adantlll 717 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 2nd `  S ) y ) )  e.  ran  ( 1st `  R ) )
8729, 5, 12rngocl 24041 . . . . . . . . . . . . . . 15  |-  ( ( R  e.  RingOps  /\  ( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) )  -> 
( ( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) )
88873expb 1189 . . . . . . . . . . . . . 14  |-  ( ( R  e.  RingOps  /\  (
( `' F `  x )  e.  ran  ( 1st `  R )  /\  ( `' F `  y )  e.  ran  ( 1st `  R ) ) )  ->  (
( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) )
8928, 88sylan2 474 . . . . . . . . . . . . 13  |-  ( ( R  e.  RingOps  /\  ( F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
)  /\  ( x  e.  ran  ( 1st `  S
)  /\  y  e.  ran  ( 1st `  S
) ) ) )  ->  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
9089anassrs 648 . . . . . . . . . . . 12  |-  ( ( ( R  e.  RingOps  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
9190adantllr 718 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) )  e. 
ran  ( 1st `  R
) )
92 f1fveq 6087 . . . . . . . . . . 11  |-  ( ( F : ran  ( 1st `  R ) -1-1-> ran  ( 1st `  S )  /\  ( ( `' F `  ( x ( 2nd `  S
) y ) )  e.  ran  ( 1st `  R )  /\  (
( `' F `  x ) ( 2nd `  R ) ( `' F `  y ) )  e.  ran  ( 1st `  R ) ) )  ->  ( ( F `  ( `' F `  ( x
( 2nd `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 2nd `  S ) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9348, 86, 91, 92syl12anc 1217 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  /\  (
x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( ( F `
 ( `' F `  ( x ( 2nd `  S ) y ) ) )  =  ( F `  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 2nd `  S ) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9493adantlrl 719 . . . . . . . . 9  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( F `  ( `' F `  ( x ( 2nd `  S
) y ) ) )  =  ( F `
 ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )  <-> 
( `' F `  ( x ( 2nd `  S ) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9581, 94mpbid 210 . . . . . . . 8  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) )
9662, 95jca 532 . . . . . . 7  |-  ( ( ( ( R  e.  RingOps 
/\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  /\  ( x  e.  ran  ( 1st `  S )  /\  y  e.  ran  ( 1st `  S ) ) )  ->  (
( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9796ralrimivva 2914 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  ->  A. x  e.  ran  ( 1st `  S ) A. y  e.  ran  ( 1st `  S ) ( ( `' F `  ( x ( 1st `  S ) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) )
9830, 7, 37, 8, 29, 5, 12, 6isrngohom 28939 . . . . . . . 8  |-  ( ( S  e.  RingOps  /\  R  e.  RingOps )  ->  ( `' F  e.  ( S  RngHom  R )  <->  ( `' F : ran  ( 1st `  S ) --> ran  ( 1st `  R )  /\  ( `' F `  (GId `  ( 2nd `  S ) ) )  =  (GId
`  ( 2nd `  R
) )  /\  A. x  e.  ran  ( 1st `  S ) A. y  e.  ran  ( 1st `  S
) ( ( `' F `  ( x ( 1st `  S
) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) ) ) )
9998ancoms 453 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( `' F  e.  ( S  RngHom  R )  <->  ( `' F : ran  ( 1st `  S ) --> ran  ( 1st `  R )  /\  ( `' F `  (GId `  ( 2nd `  S ) ) )  =  (GId
`  ( 2nd `  R
) )  /\  A. x  e.  ran  ( 1st `  S ) A. y  e.  ran  ( 1st `  S
) ( ( `' F `  ( x ( 1st `  S
) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) ) ) )
10099adantr 465 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( `' F  e.  ( S  RngHom  R )  <-> 
( `' F : ran  ( 1st `  S
) --> ran  ( 1st `  R )  /\  ( `' F `  (GId `  ( 2nd `  S ) ) )  =  (GId
`  ( 2nd `  R
) )  /\  A. x  e.  ran  ( 1st `  S ) A. y  e.  ran  ( 1st `  S
) ( ( `' F `  ( x ( 1st `  S
) y ) )  =  ( ( `' F `  x ) ( 1st `  R
) ( `' F `  y ) )  /\  ( `' F `  ( x ( 2nd `  S
) y ) )  =  ( ( `' F `  x ) ( 2nd `  R
) ( `' F `  y ) ) ) ) ) )
1014, 18, 97, 100mpbir3and 1171 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  ->  `' F  e.  ( S  RngHom  R ) )
1021ad2antll 728 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  ->  `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R ) )
103101, 102jca 532 . . . 4  |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) )  -> 
( `' F  e.  ( S  RngHom  R )  /\  `' F : ran  ( 1st `  S
)
-1-1-onto-> ran  ( 1st `  R
) ) )
104103ex 434 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  (
( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) )  ->  ( `' F  e.  ( S  RngHom  R )  /\  `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R ) ) ) )
10529, 12, 30, 37isrngoiso 28952 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  <->  ( F  e.  ( R  RngHom  S )  /\  F : ran  ( 1st `  R ) -1-1-onto-> ran  ( 1st `  S
) ) ) )
10630, 37, 29, 12isrngoiso 28952 . . . 4  |-  ( ( S  e.  RingOps  /\  R  e.  RingOps )  ->  ( `' F  e.  ( S  RngIso  R )  <->  ( `' F  e.  ( S  RngHom  R )  /\  `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R
) ) ) )
107106ancoms 453 . . 3  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( `' F  e.  ( S  RngIso  R )  <->  ( `' F  e.  ( S  RngHom  R )  /\  `' F : ran  ( 1st `  S ) -1-1-onto-> ran  ( 1st `  R
) ) ) )
108104, 105, 1073imtr4d 268 . 2  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  ->  `' F  e.  ( S  RngIso  R ) ) )
1091083impia 1185 1  |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  `' F  e.  ( S  RngIso  R ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   `'ccnv 4950   ran crn 4952   -->wf 5525   -1-1->wf1 5526   -1-1-onto->wf1o 5528   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689  GIdcgi 23846   RingOpscrngo 24034    RngHom crnghom 28934    RngIso crngiso 28935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-map 7329  df-grpo 23850  df-gid 23851  df-ablo 23941  df-ass 23972  df-exid 23974  df-mgm 23978  df-sgr 23990  df-mndo 23997  df-rngo 24035  df-rngohom 28937  df-rngoiso 28950
This theorem is referenced by:  riscer  28962
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