MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isoeq1 Structured version   Unicode version

Theorem isoeq1 6112
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 5733 . . 3  |-  ( H  =  G  ->  ( H : A -1-1-onto-> B  <->  G : A -1-1-onto-> B ) )
2 fveq1 5791 . . . . . 6  |-  ( H  =  G  ->  ( H `  x )  =  ( G `  x ) )
3 fveq1 5791 . . . . . 6  |-  ( H  =  G  ->  ( H `  y )  =  ( G `  y ) )
42, 3breq12d 4406 . . . . 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 2871 . . 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 5528 . 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 5528 . 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 1370   A.wral 2795   class class class wbr 4393   -1-1-onto->wf1o 5518   ` cfv 5519    Isom wiso 5520
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-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
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-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528
This theorem is referenced by:  isores1  6127  wemoiso  6665  wemoiso2  6666  ordiso  7834  oieu  7857  finnisoeu  8387  iunfictbso  8388  infmsup  10412  ltweuz  11894  fz1isolem  12325  isercolllem2  13254  isercoll  13256  dvgt0lem2  21601  efcvx  22040  relogiso  22172  logccv  22234  erdszelem1  27216  erdsze  27227  erdsze2lem2  27229
  Copyright terms: Public domain W3C validator