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Theorem isoeq1 6200
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 5797 . . 3  |-  ( H  =  G  ->  ( H : A -1-1-onto-> B  <->  G : A -1-1-onto-> B ) )
2 fveq1 5855 . . . . . 6  |-  ( H  =  G  ->  ( H `  x )  =  ( G `  x ) )
3 fveq1 5855 . . . . . 6  |-  ( H  =  G  ->  ( H `  y )  =  ( G `  y ) )
42, 3breq12d 4450 . . . . 5  |-  ( H  =  G  ->  (
( H `  x
) S ( H `
 y )  <->  ( G `  x ) S ( G `  y ) ) )
54bibi2d 318 . . . 4  |-  ( H  =  G  ->  (
( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <-> 
( x R y  <-> 
( G `  x
) S ( G `
 y ) ) ) )
652ralbidv 2887 . . 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 710 . 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 5587 . 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 5587 . 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 288 1  |-  ( H  =  G  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
G  Isom  R ,  S  ( A ,  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383   A.wral 2793   class class class wbr 4437   -1-1-onto->wf1o 5577   ` cfv 5578    Isom wiso 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587
This theorem is referenced by:  isores1  6215  wemoiso  6770  wemoiso2  6771  ordiso  7944  oieu  7967  finnisoeu  8497  iunfictbso  8498  infmsup  10528  ltweuz  12054  fz1isolem  12492  isercolllem2  13470  isercoll  13472  dvgt0lem2  22382  efcvx  22822  relogiso  22960  logccv  23022  erdszelem1  28613  erdsze  28624  erdsze2lem2  28626  fzisoeu  31454  fourierdlem36  31879  fourierdlem96  31939  fourierdlem97  31940  fourierdlem98  31941  fourierdlem99  31942  fourierdlem105  31948  fourierdlem106  31949  fourierdlem108  31951  fourierdlem110  31953  fourierdlem112  31955  fourierdlem113  31956  fourierdlem115  31958
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