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Theorem isoini 5687
Description: Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
Assertion
Ref Expression
isoini  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  ( H " ( A  i^i  ( `' R " { D } ) ) )  =  ( B  i^i  ( `' S " { ( H `  D ) } ) ) )

Proof of Theorem isoini
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3229 . . . 4  |-  ( y  e.  ( B  i^i  ( `' S " { ( H `  D ) } ) )  <->  ( y  e.  B  /\  y  e.  ( `' S " { ( H `  D ) } ) ) )
2 isof1o 5676 . . . . . . . . 9  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
3 f1ofo 5342 . . . . . . . . 9  |-  ( H : A -1-1-onto-> B  ->  H : A -onto-> B )
4 forn 5318 . . . . . . . . . 10  |-  ( H : A -onto-> B  ->  ran  H  =  B )
54eleq2d 2187 . . . . . . . . 9  |-  ( H : A -onto-> B  -> 
( y  e.  ran  H  <-> 
y  e.  B ) )
62, 3, 53syl 17 . . . . . . . 8  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( y  e. 
ran  H  <->  y  e.  B
) )
7 f1ofn 5336 . . . . . . . . 9  |-  ( H : A -1-1-onto-> B  ->  H  Fn  A )
8 fvelrnb 5437 . . . . . . . . 9  |-  ( H  Fn  A  ->  (
y  e.  ran  H  <->  E. x  e.  A  ( H `  x )  =  y ) )
92, 7, 83syl 17 . . . . . . . 8  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( y  e. 
ran  H  <->  E. x  e.  A  ( H `  x )  =  y ) )
106, 9bitr3d 189 . . . . . . 7  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( y  e.  B  <->  E. x  e.  A  ( H `  x )  =  y ) )
1110adantr 274 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  (
y  e.  B  <->  E. x  e.  A  ( H `  x )  =  y ) )
122, 7syl 14 . . . . . . . 8  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Fn  A
)
1312anim1i 338 . . . . . . 7  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  ( H  Fn  A  /\  D  e.  A )
)
14 funfvex 5406 . . . . . . . 8  |-  ( ( Fun  H  /\  D  e.  dom  H )  -> 
( H `  D
)  e.  _V )
1514funfni 5193 . . . . . . 7  |-  ( ( H  Fn  A  /\  D  e.  A )  ->  ( H `  D
)  e.  _V )
16 vex 2663 . . . . . . . 8  |-  y  e. 
_V
1716eliniseg 4879 . . . . . . 7  |-  ( ( H `  D )  e.  _V  ->  (
y  e.  ( `' S " { ( H `  D ) } )  <->  y S
( H `  D
) ) )
1813, 15, 173syl 17 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  (
y  e.  ( `' S " { ( H `  D ) } )  <->  y S
( H `  D
) ) )
1911, 18anbi12d 464 . . . . 5  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  (
( y  e.  B  /\  y  e.  ( `' S " { ( H `  D ) } ) )  <->  ( E. x  e.  A  ( H `  x )  =  y  /\  y S ( H `  D ) ) ) )
20 elin 3229 . . . . . . . . . . . 12  |-  ( x  e.  ( A  i^i  ( `' R " { D } ) )  <->  ( x  e.  A  /\  x  e.  ( `' R " { D } ) ) )
21 vex 2663 . . . . . . . . . . . . . 14  |-  x  e. 
_V
2221eliniseg 4879 . . . . . . . . . . . . 13  |-  ( D  e.  A  ->  (
x  e.  ( `' R " { D } )  <->  x R D ) )
2322anbi2d 459 . . . . . . . . . . . 12  |-  ( D  e.  A  ->  (
( x  e.  A  /\  x  e.  ( `' R " { D } ) )  <->  ( x  e.  A  /\  x R D ) ) )
2420, 23syl5bb 191 . . . . . . . . . . 11  |-  ( D  e.  A  ->  (
x  e.  ( A  i^i  ( `' R " { D } ) )  <->  ( x  e.  A  /\  x R D ) ) )
2524anbi1d 460 . . . . . . . . . 10  |-  ( D  e.  A  ->  (
( x  e.  ( A  i^i  ( `' R " { D } ) )  /\  x H y )  <->  ( (
x  e.  A  /\  x R D )  /\  x H y ) ) )
26 anass 398 . . . . . . . . . 10  |-  ( ( ( x  e.  A  /\  x R D )  /\  x H y )  <->  ( x  e.  A  /\  ( x R D  /\  x H y ) ) )
2725, 26syl6bb 195 . . . . . . . . 9  |-  ( D  e.  A  ->  (
( x  e.  ( A  i^i  ( `' R " { D } ) )  /\  x H y )  <->  ( x  e.  A  /\  (
x R D  /\  x H y ) ) ) )
2827adantl 275 . . . . . . . 8  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  (
( x  e.  ( A  i^i  ( `' R " { D } ) )  /\  x H y )  <->  ( x  e.  A  /\  (
x R D  /\  x H y ) ) ) )
29 isorel 5677 . . . . . . . . . . . . . 14  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x R D  <->  ( H `  x ) S ( H `  D ) ) )
30 fnbrfvb 5430 . . . . . . . . . . . . . . . . 17  |-  ( ( H  Fn  A  /\  x  e.  A )  ->  ( ( H `  x )  =  y  <-> 
x H y ) )
3130bicomd 140 . . . . . . . . . . . . . . . 16  |-  ( ( H  Fn  A  /\  x  e.  A )  ->  ( x H y  <-> 
( H `  x
)  =  y ) )
3212, 31sylan 281 . . . . . . . . . . . . . . 15  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  x  e.  A )  ->  (
x H y  <->  ( H `  x )  =  y ) )
3332adantrr 470 . . . . . . . . . . . . . 14  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x H y  <->  ( H `  x )  =  y ) )
3429, 33anbi12d 464 . . . . . . . . . . . . 13  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( (
x R D  /\  x H y )  <->  ( ( H `  x ) S ( H `  D )  /\  ( H `  x )  =  y ) ) )
35 ancom 264 . . . . . . . . . . . . . 14  |-  ( ( ( H `  x
) S ( H `
 D )  /\  ( H `  x )  =  y )  <->  ( ( H `  x )  =  y  /\  ( H `  x ) S ( H `  D ) ) )
36 breq1 3902 . . . . . . . . . . . . . . 15  |-  ( ( H `  x )  =  y  ->  (
( H `  x
) S ( H `
 D )  <->  y S
( H `  D
) ) )
3736pm5.32i 449 . . . . . . . . . . . . . 14  |-  ( ( ( H `  x
)  =  y  /\  ( H `  x ) S ( H `  D ) )  <->  ( ( H `  x )  =  y  /\  y S ( H `  D ) ) )
3835, 37bitri 183 . . . . . . . . . . . . 13  |-  ( ( ( H `  x
) S ( H `
 D )  /\  ( H `  x )  =  y )  <->  ( ( H `  x )  =  y  /\  y S ( H `  D ) ) )
3934, 38syl6bb 195 . . . . . . . . . . . 12  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( (
x R D  /\  x H y )  <->  ( ( H `  x )  =  y  /\  y S ( H `  D ) ) ) )
4039exp32 362 . . . . . . . . . . 11  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( x  e.  A  ->  ( D  e.  A  ->  ( ( x R D  /\  x H y )  <->  ( ( H `  x )  =  y  /\  y S ( H `  D ) ) ) ) ) )
4140com23 78 . . . . . . . . . 10  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( D  e.  A  ->  ( x  e.  A  ->  ( ( x R D  /\  x H y )  <->  ( ( H `  x )  =  y  /\  y S ( H `  D ) ) ) ) ) )
4241imp 123 . . . . . . . . 9  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  (
x  e.  A  -> 
( ( x R D  /\  x H y )  <->  ( ( H `  x )  =  y  /\  y S ( H `  D ) ) ) ) )
4342pm5.32d 445 . . . . . . . 8  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  (
( x  e.  A  /\  ( x R D  /\  x H y ) )  <->  ( x  e.  A  /\  (
( H `  x
)  =  y  /\  y S ( H `  D ) ) ) ) )
4428, 43bitrd 187 . . . . . . 7  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  (
( x  e.  ( A  i^i  ( `' R " { D } ) )  /\  x H y )  <->  ( x  e.  A  /\  (
( H `  x
)  =  y  /\  y S ( H `  D ) ) ) ) )
4544rexbidv2 2417 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  ( E. x  e.  ( A  i^i  ( `' R " { D } ) ) x H y  <->  E. x  e.  A  ( ( H `  x )  =  y  /\  y S ( H `  D ) ) ) )
46 r19.41v 2564 . . . . . 6  |-  ( E. x  e.  A  ( ( H `  x
)  =  y  /\  y S ( H `  D ) )  <->  ( E. x  e.  A  ( H `  x )  =  y  /\  y S ( H `  D ) ) )
4745, 46syl6bb 195 . . . . 5  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  ( E. x  e.  ( A  i^i  ( `' R " { D } ) ) x H y  <-> 
( E. x  e.  A  ( H `  x )  =  y  /\  y S ( H `  D ) ) ) )
4819, 47bitr4d 190 . . . 4  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  (
( y  e.  B  /\  y  e.  ( `' S " { ( H `  D ) } ) )  <->  E. x  e.  ( A  i^i  ( `' R " { D } ) ) x H y ) )
491, 48syl5bb 191 . . 3  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  (
y  e.  ( B  i^i  ( `' S " { ( H `  D ) } ) )  <->  E. x  e.  ( A  i^i  ( `' R " { D } ) ) x H y ) )
5049abbi2dv 2236 . 2  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  ( B  i^i  ( `' S " { ( H `  D ) } ) )  =  { y  |  E. x  e.  ( A  i^i  ( `' R " { D } ) ) x H y } )
51 dfima2 4853 . 2  |-  ( H
" ( A  i^i  ( `' R " { D } ) ) )  =  { y  |  E. x  e.  ( A  i^i  ( `' R " { D } ) ) x H y }
5250, 51syl6reqr 2169 1  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  D  e.  A )  ->  ( H " ( A  i^i  ( `' R " { D } ) ) )  =  ( B  i^i  ( `' S " { ( H `  D ) } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   {cab 2103   E.wrex 2394   _Vcvv 2660    i^i cin 3040   {csn 3497   class class class wbr 3899   `'ccnv 4508   ran crn 4510   "cima 4512    Fn wfn 5088   -onto->wfo 5091   -1-1-onto->wf1o 5092   ` cfv 5093    Isom wiso 5094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-isom 5102
This theorem is referenced by:  isoini2  5688  isoselem  5689
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