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Theorem isoeq1 5466
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 5142 . . 3  |-  ( H  =  G  ->  ( H : A -1-1-onto-> B  <->  G : A -1-1-onto-> B ) )
2 fveq1 5202 . . . . . 6  |-  ( H  =  G  ->  ( H `  x )  =  ( G `  x ) )
3 fveq1 5202 . . . . . 6  |-  ( H  =  G  ->  ( H `  y )  =  ( G `  y ) )
42, 3breq12d 3800 . . . . 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 2391 . . 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 4935 . 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 4935 . 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 1285   A.wral 2349   class class class wbr 3787   -1-1-onto->wf1o 4925   ` cfv 4926    Isom wiso 4927
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-isom 4935
This theorem is referenced by:  isores1  5479  ordiso  6496  infrenegsupex  8752
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