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Theorem isoini 6050
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 3522 . . . 4  |-  ( y  e.  ( B  i^i  ( `' S " { ( H `  D ) } ) )  <->  ( y  e.  B  /\  y  e.  ( `' S " { ( H `  D ) } ) ) )
2 isof1o 6037 . . . . . . . . 9  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
3 f1ofo 5673 . . . . . . . . 9  |-  ( H : A -1-1-onto-> B  ->  H : A -onto-> B )
4 forn 5648 . . . . . . . . . 10  |-  ( H : A -onto-> B  ->  ran  H  =  B )
54eleq2d 2502 . . . . . . . . 9  |-  ( H : A -onto-> B  -> 
( y  e.  ran  H  <-> 
y  e.  B ) )
62, 3, 53syl 19 . . . . . . . 8  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( y  e. 
ran  H  <->  y  e.  B
) )
7 f1ofn 5667 . . . . . . . . 9  |-  ( H : A -1-1-onto-> B  ->  H  Fn  A )
8 fvelrnb 5766 . . . . . . . . 9  |-  ( H  Fn  A  ->  (
y  e.  ran  H  <->  E. x  e.  A  ( H `  x )  =  y ) )
92, 7, 83syl 19 . . . . . . . 8  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( y  e. 
ran  H  <->  E. x  e.  A  ( H `  x )  =  y ) )
106, 9bitr3d 247 . . . . . . 7  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( y  e.  B  <->  E. x  e.  A  ( H `  x )  =  y ) )
11 fvex 5734 . . . . . . . 8  |-  ( H `
 D )  e. 
_V
12 vex 2951 . . . . . . . . 9  |-  y  e. 
_V
1312eliniseg 5225 . . . . . . . 8  |-  ( ( H `  D )  e.  _V  ->  (
y  e.  ( `' S " { ( H `  D ) } )  <->  y S
( H `  D
) ) )
1411, 13mp1i 12 . . . . . . 7  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( y  e.  ( `' S " { ( H `  D ) } )  <-> 
y S ( H `
 D ) ) )
1510, 14anbi12d 692 . . . . . 6  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( ( y  e.  B  /\  y  e.  ( `' S " { ( H `  D ) } ) )  <->  ( E. x  e.  A  ( H `  x )  =  y  /\  y S ( H `  D ) ) ) )
1615adantr 452 . . . . 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 ) ) ) )
17 elin 3522 . . . . . . . . . . . 12  |-  ( x  e.  ( A  i^i  ( `' R " { D } ) )  <->  ( x  e.  A  /\  x  e.  ( `' R " { D } ) ) )
18 vex 2951 . . . . . . . . . . . . . 14  |-  x  e. 
_V
1918eliniseg 5225 . . . . . . . . . . . . 13  |-  ( D  e.  A  ->  (
x  e.  ( `' R " { D } )  <->  x R D ) )
2019anbi2d 685 . . . . . . . . . . . 12  |-  ( D  e.  A  ->  (
( x  e.  A  /\  x  e.  ( `' R " { D } ) )  <->  ( x  e.  A  /\  x R D ) ) )
2117, 20syl5bb 249 . . . . . . . . . . 11  |-  ( D  e.  A  ->  (
x  e.  ( A  i^i  ( `' R " { D } ) )  <->  ( x  e.  A  /\  x R D ) ) )
2221anbi1d 686 . . . . . . . . . 10  |-  ( D  e.  A  ->  (
( x  e.  ( A  i^i  ( `' R " { D } ) )  /\  x H y )  <->  ( (
x  e.  A  /\  x R D )  /\  x H y ) ) )
23 anass 631 . . . . . . . . . 10  |-  ( ( ( x  e.  A  /\  x R D )  /\  x H y )  <->  ( x  e.  A  /\  ( x R D  /\  x H y ) ) )
2422, 23syl6bb 253 . . . . . . . . 9  |-  ( D  e.  A  ->  (
( x  e.  ( A  i^i  ( `' R " { D } ) )  /\  x H y )  <->  ( x  e.  A  /\  (
x R D  /\  x H y ) ) ) )
2524adantl 453 . . . . . . . 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 ) ) ) )
26 isorel 6038 . . . . . . . . . . . . . 14  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x R D  <->  ( H `  x ) S ( H `  D ) ) )
272, 7syl 16 . . . . . . . . . . . . . . . 16  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Fn  A
)
28 fnbrfvb 5759 . . . . . . . . . . . . . . . . 17  |-  ( ( H  Fn  A  /\  x  e.  A )  ->  ( ( H `  x )  =  y  <-> 
x H y ) )
2928bicomd 193 . . . . . . . . . . . . . . . 16  |-  ( ( H  Fn  A  /\  x  e.  A )  ->  ( x H y  <-> 
( H `  x
)  =  y ) )
3027, 29sylan 458 . . . . . . . . . . . . . . 15  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  x  e.  A )  ->  (
x H y  <->  ( H `  x )  =  y ) )
3130adantrr 698 . . . . . . . . . . . . . 14  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x H y  <->  ( H `  x )  =  y ) )
3226, 31anbi12d 692 . . . . . . . . . . . . 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 ) ) )
33 ancom 438 . . . . . . . . . . . . . 14  |-  ( ( ( H `  x
) S ( H `
 D )  /\  ( H `  x )  =  y )  <->  ( ( H `  x )  =  y  /\  ( H `  x ) S ( H `  D ) ) )
34 breq1 4207 . . . . . . . . . . . . . . 15  |-  ( ( H `  x )  =  y  ->  (
( H `  x
) S ( H `
 D )  <->  y S
( H `  D
) ) )
3534pm5.32i 619 . . . . . . . . . . . . . 14  |-  ( ( ( H `  x
)  =  y  /\  ( H `  x ) S ( H `  D ) )  <->  ( ( H `  x )  =  y  /\  y S ( H `  D ) ) )
3633, 35bitri 241 . . . . . . . . . . . . 13  |-  ( ( ( H `  x
) S ( H `
 D )  /\  ( H `  x )  =  y )  <->  ( ( H `  x )  =  y  /\  y S ( H `  D ) ) )
3732, 36syl6bb 253 . . . . . . . . . . . 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 ) ) ) )
3837exp32 589 . . . . . . . . . . 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 ) ) ) ) ) )
3938com23 74 . . . . . . . . . 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 ) ) ) ) ) )
4039imp 419 . . . . . . . . 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 ) ) ) ) )
4140pm5.32d 621 . . . . . . . 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 ) ) ) ) )
4225, 41bitrd 245 . . . . . . 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 ) ) ) ) )
4342rexbidv2 2720 . . . . . 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 ) ) ) )
44 r19.41v 2853 . . . . . 6  |-  ( E. x  e.  A  ( ( H `  x
)  =  y  /\  y S ( H `  D ) )  <->  ( E. x  e.  A  ( H `  x )  =  y  /\  y S ( H `  D ) ) )
4543, 44syl6bb 253 . . . . 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 ) ) ) )
4616, 45bitr4d 248 . . . 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 ) )
471, 46syl5bb 249 . . 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 ) )
4847abbi2dv 2550 . 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 } )
49 dfima2 5197 . 2  |-  ( H
" ( A  i^i  ( `' R " { D } ) ) )  =  { y  |  E. x  e.  ( A  i^i  ( `' R " { D } ) ) x H y }
5048, 49syl6reqr 2486 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    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   _Vcvv 2948    i^i cin 3311   {csn 3806   class class class wbr 4204   `'ccnv 4869   ran crn 4871   "cima 4873    Fn wfn 5441   -onto->wfo 5444   -1-1-onto->wf1o 5445   ` cfv 5446    Isom wiso 5447
This theorem is referenced by:  isoini2  6051  isoselem  6053  infxpenlem  7887
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455
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