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Theorem isoeq1 5580
Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
Assertion
Ref Expression
isoeq1  |-  ( H  =  G  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
G  Isom  R ,  S  ( A ,  B ) ) )

Proof of Theorem isoeq1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oeq1 5244 . . 3  |-  ( H  =  G  ->  ( H : A -1-1-onto-> B  <->  G : A -1-1-onto-> B ) )
2 fveq1 5304 . . . . . 6  |-  ( H  =  G  ->  ( H `  x )  =  ( G `  x ) )
3 fveq1 5304 . . . . . 6  |-  ( H  =  G  ->  ( H `  y )  =  ( G `  y ) )
42, 3breq12d 3858 . . . . 5  |-  ( H  =  G  ->  (
( H `  x
) S ( H `
 y )  <->  ( G `  x ) S ( G `  y ) ) )
54bibi2d 230 . . . 4  |-  ( H  =  G  ->  (
( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <-> 
( x R y  <-> 
( G `  x
) S ( G `
 y ) ) ) )
652ralbidv 2402 . . 3  |-  ( H  =  G  ->  ( A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <->  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( G `  x
) S ( G `
 y ) ) ) )
71, 6anbi12d 457 . 2  |-  ( H  =  G  ->  (
( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) )  <->  ( G : A
-1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <->  ( G `  x ) S ( G `  y ) ) ) ) )
8 df-isom 5024 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
9 df-isom 5024 . 2  |-  ( G 
Isom  R ,  S  ( A ,  B )  <-> 
( G : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( G `  x
) S ( G `
 y ) ) ) )
107, 8, 93bitr4g 221 1  |-  ( H  =  G  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
G  Isom  R ,  S  ( A ,  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   A.wral 2359   class class class wbr 3845   -1-1-onto->wf1o 5014   ` cfv 5015    Isom wiso 5016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-isom 5024
This theorem is referenced by:  isores1  5593  ordiso  6729  infrenegsupex  9082  zfz1isolem1  10245  zfz1iso  10246
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