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Theorem isores2 5584
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 5247 . . . . . . . 8  |-  ( H : A -1-1-onto-> B  ->  H : A
--> B )
2 ffvelrn 5426 . . . . . . . . . 10  |-  ( ( H : A --> B  /\  x  e.  A )  ->  ( H `  x
)  e.  B )
32adantrr 463 . . . . . . . . 9  |-  ( ( H : A --> B  /\  ( x  e.  A  /\  y  e.  A
) )  ->  ( H `  x )  e.  B )
4 ffvelrn 5426 . . . . . . . . . 10  |-  ( ( H : A --> B  /\  y  e.  A )  ->  ( H `  y
)  e.  B )
54adantrl 462 . . . . . . . . 9  |-  ( ( H : A --> B  /\  ( x  e.  A  /\  y  e.  A
) )  ->  ( H `  y )  e.  B )
6 brinxp 4502 . . . . . . . . 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 403 . . . . . . . 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 277 . . . . . . 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 392 . . . . . 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 230 . . . . 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 2376 . . . 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 2376 . . 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 442 . 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 5019 . 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 5019 . 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 210 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:    /\ wa 102    <-> wb 103    e. wcel 1438   A.wral 2359    i^i cin 2998   class class class wbr 3843    X. cxp 4434   -->wf 5006   -1-1-onto->wf1o 5009   ` cfv 5010    Isom wiso 5011
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-id 4118  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-f1o 5017  df-fv 5018  df-isom 5019
This theorem is referenced by:  isores1  5585
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