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Theorem isores2 5811
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 5460 . . . . . . . 8  |-  ( H : A -1-1-onto-> B  ->  H : A
--> B )
2 ffvelcdm 5648 . . . . . . . . . 10  |-  ( ( H : A --> B  /\  x  e.  A )  ->  ( H `  x
)  e.  B )
32adantrr 479 . . . . . . . . 9  |-  ( ( H : A --> B  /\  ( x  e.  A  /\  y  e.  A
) )  ->  ( H `  x )  e.  B )
4 ffvelcdm 5648 . . . . . . . . . 10  |-  ( ( H : A --> B  /\  y  e.  A )  ->  ( H `  y
)  e.  B )
54adantrl 478 . . . . . . . . 9  |-  ( ( H : A --> B  /\  ( x  e.  A  /\  y  e.  A
) )  ->  ( H `  y )  e.  B )
6 brinxp 4693 . . . . . . . . 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 411 . . . . . . . 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 283 . . . . . . 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 400 . . . . . 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 232 . . . . 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 2473 . . . 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 2473 . . 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 454 . 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 5224 . 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 5224 . 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 212 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 104    <-> wb 105    e. wcel 2148   A.wral 2455    i^i cin 3128   class class class wbr 4002    X. cxp 4623   -->wf 5211   -1-1-onto->wf1o 5214   ` cfv 5215    Isom wiso 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-f1o 5222  df-fv 5223  df-isom 5224
This theorem is referenced by:  isores1  5812
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