MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isores2 Structured version   Unicode version

Theorem isores2 6053
Description: An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)
Assertion
Ref Expression
isores2  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R , 
( S  i^i  ( B  X.  B ) ) ( A ,  B
) )

Proof of Theorem isores2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1of 5674 . . . . . . . 8  |-  ( H : A -1-1-onto-> B  ->  H : A
--> B )
2 ffvelrn 5868 . . . . . . . . . 10  |-  ( ( H : A --> B  /\  x  e.  A )  ->  ( H `  x
)  e.  B )
32adantrr 698 . . . . . . . . 9  |-  ( ( H : A --> B  /\  ( x  e.  A  /\  y  e.  A
) )  ->  ( H `  x )  e.  B )
4 ffvelrn 5868 . . . . . . . . . 10  |-  ( ( H : A --> B  /\  y  e.  A )  ->  ( H `  y
)  e.  B )
54adantrl 697 . . . . . . . . 9  |-  ( ( H : A --> B  /\  ( x  e.  A  /\  y  e.  A
) )  ->  ( H `  y )  e.  B )
6 brinxp 4940 . . . . . . . . 9  |-  ( ( ( H `  x
)  e.  B  /\  ( H `  y )  e.  B )  -> 
( ( H `  x ) S ( H `  y )  <-> 
( H `  x
) ( S  i^i  ( B  X.  B
) ) ( H `
 y ) ) )
73, 5, 6syl2anc 643 . . . . . . . 8  |-  ( ( H : A --> B  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
( H `  x
) S ( H `
 y )  <->  ( H `  x ) ( S  i^i  ( B  X.  B ) ) ( H `  y ) ) )
81, 7sylan 458 . . . . . . 7  |-  ( ( H : A -1-1-onto-> B  /\  ( x  e.  A  /\  y  e.  A
) )  ->  (
( H `  x
) S ( H `
 y )  <->  ( H `  x ) ( S  i^i  ( B  X.  B ) ) ( H `  y ) ) )
98anassrs 630 . . . . . 6  |-  ( ( ( H : A -1-1-onto-> B  /\  x  e.  A
)  /\  y  e.  A )  ->  (
( H `  x
) S ( H `
 y )  <->  ( H `  x ) ( S  i^i  ( B  X.  B ) ) ( H `  y ) ) )
109bibi2d 310 . . . . 5  |-  ( ( ( H : A -1-1-onto-> B  /\  x  e.  A
)  /\  y  e.  A )  ->  (
( x R y  <-> 
( H `  x
) S ( H `
 y ) )  <-> 
( x R y  <-> 
( H `  x
) ( S  i^i  ( B  X.  B
) ) ( H `
 y ) ) ) )
1110ralbidva 2721 . . . 4  |-  ( ( H : A -1-1-onto-> B  /\  x  e.  A )  ->  ( A. y  e.  A  ( x R y  <->  ( H `  x ) S ( H `  y ) )  <->  A. y  e.  A  ( x R y  <-> 
( H `  x
) ( S  i^i  ( B  X.  B
) ) ( H `
 y ) ) ) )
1211ralbidva 2721 . . 3  |-  ( H : A -1-1-onto-> B  ->  ( 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  <-> 
( H `  x
) ( S  i^i  ( B  X.  B
) ) ( H `
 y ) ) ) )
1312pm5.32i 619 . 2  |-  ( ( 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-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <->  ( H `  x ) ( S  i^i  ( B  X.  B ) ) ( H `  y ) ) ) )
14 df-isom 5463 . 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 ) ) ) )
15 df-isom 5463 . 2  |-  ( H 
Isom  R ,  ( S  i^i  ( B  X.  B ) ) ( A ,  B )  <-> 
( H : A -1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x R y  <-> 
( H `  x
) ( S  i^i  ( B  X.  B
) ) ( H `
 y ) ) ) )
1613, 14, 153bitr4i 269 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  R , 
( S  i^i  ( B  X.  B ) ) ( A ,  B
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1725   A.wral 2705    i^i cin 3319   class class class wbr 4212    X. cxp 4876   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454    Isom wiso 5455
This theorem is referenced by:  isores1  6054  hartogslem1  7511  leiso  11708  icopnfhmeo  18968  iccpnfhmeo  18970  gtiso  24088  xrge0iifhmeo  24322
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-f1o 5461  df-fv 5462  df-isom 5463
  Copyright terms: Public domain W3C validator