MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isoeq5 Structured version   Unicode version

Theorem isoeq5 6099
Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
Assertion
Ref Expression
isoeq5  |-  ( B  =  C  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R ,  S  ( A ,  C ) ) )

Proof of Theorem isoeq5
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oeq3 5718 . . 3  |-  ( B  =  C  ->  ( H : A -1-1-onto-> B  <->  H : A -1-1-onto-> C ) )
21anbi1d 704 . 2  |-  ( B  =  C  ->  (
( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) )  <->  ( H : A
-1-1-onto-> C  /\  A. x  e.  A  A. y  e.  A  ( x R y  <->  ( H `  x ) S ( H `  y ) ) ) ) )
3 df-isom 5511 . 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 ) ) ) )
4 df-isom 5511 . 2  |-  ( H 
Isom  R ,  S  ( A ,  C )  <-> 
( H : A -1-1-onto-> C  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) S ( H `
 y ) ) ) )
52, 3, 43bitr4g 288 1  |-  ( B  =  C  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R ,  S  ( A ,  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   A.wral 2792   class class class wbr 4376   -1-1-onto->wf1o 5501   ` cfv 5502    Isom wiso 5503
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 1709  ax-7 1729  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-in 3419  df-ss 3426  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-isom 5511
This theorem is referenced by:  isores3  6111  ordiso  7817  ordtypelem9  7827  ordtypelem10  7828  oiid  7842  iunfictbso  8371  ltweuz  11871  fz1isolem  12302  dvgt0lem2  21577  erdszelem1  27199  erdsze  27210  erdsze2lem1  27211  erdsze2lem2  27212
  Copyright terms: Public domain W3C validator