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Theorem isomin 5850
Description: Isomorphisms preserve minimal elements. Note that  ( `' R " { D } ) is Takeuti and Zaring's idiom for the initial segment  { x  |  x R D }. Proposition 6.31(1) of [TakeutiZaring] p. 33. (Contributed by NM, 19-Apr-2004.)
Assertion
Ref Expression
isomin  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( ( C  i^i  ( `' R " { D } ) )  =  (/) 
<->  ( ( H " C )  i^i  ( `' S " { ( H `  D ) } ) )  =  (/) ) )

Proof of Theorem isomin
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neq0 3478 . . . 4  |-  ( -.  ( ( H " C )  i^i  ( `' S " { ( H `  D ) } ) )  =  (/) 
<->  E. y  y  e.  ( ( H " C )  i^i  ( `' S " { ( H `  D ) } ) ) )
2 ssel 3187 . . . . . . . . . . . . . 14  |-  ( C 
C_  A  ->  (
x  e.  C  ->  x  e.  A )
)
3 isof1o 5838 . . . . . . . . . . . . . . 15  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
4 f1ofn 5489 . . . . . . . . . . . . . . 15  |-  ( H : A -1-1-onto-> B  ->  H  Fn  A )
5 fnbrfvb 5579 . . . . . . . . . . . . . . . 16  |-  ( ( H  Fn  A  /\  x  e.  A )  ->  ( ( H `  x )  =  y  <-> 
x H y ) )
65ex 423 . . . . . . . . . . . . . . 15  |-  ( H  Fn  A  ->  (
x  e.  A  -> 
( ( H `  x )  =  y  <-> 
x H y ) ) )
73, 4, 63syl 18 . . . . . . . . . . . . . 14  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( x  e.  A  ->  ( ( H `  x )  =  y  <->  x H y ) ) )
82, 7syl9r 67 . . . . . . . . . . . . 13  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( C  C_  A  ->  ( x  e.  C  ->  ( ( H `  x )  =  y  <->  x H y ) ) ) )
98imp31 421 . . . . . . . . . . . 12  |-  ( ( ( H  Isom  R ,  S  ( A ,  B )  /\  C  C_  A )  /\  x  e.  C )  ->  (
( H `  x
)  =  y  <->  x H
y ) )
109rexbidva 2573 . . . . . . . . . . 11  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  C  C_  A )  ->  ( E. x  e.  C  ( H `  x )  =  y  <->  E. x  e.  C  x H
y ) )
11 vex 2804 . . . . . . . . . . . 12  |-  y  e. 
_V
1211elima 5033 . . . . . . . . . . 11  |-  ( y  e.  ( H " C )  <->  E. x  e.  C  x H
y )
1310, 12syl6rbbr 255 . . . . . . . . . 10  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  C  C_  A )  ->  (
y  e.  ( H
" C )  <->  E. x  e.  C  ( H `  x )  =  y ) )
14 fvex 5555 . . . . . . . . . . 11  |-  ( H `
 D )  e. 
_V
1511eliniseg 5058 . . . . . . . . . . 11  |-  ( ( H `  D )  e.  _V  ->  (
y  e.  ( `' S " { ( H `  D ) } )  <->  y S
( H `  D
) ) )
1614, 15mp1i 11 . . . . . . . . . 10  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  C  C_  A )  ->  (
y  e.  ( `' S " { ( H `  D ) } )  <->  y S
( H `  D
) ) )
1713, 16anbi12d 691 . . . . . . . . 9  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  C  C_  A )  ->  (
( y  e.  ( H " C )  /\  y  e.  ( `' S " { ( H `  D ) } ) )  <->  ( E. x  e.  C  ( H `  x )  =  y  /\  y S ( H `  D ) ) ) )
18 elin 3371 . . . . . . . . 9  |-  ( y  e.  ( ( H
" C )  i^i  ( `' S " { ( H `  D ) } ) )  <->  ( y  e.  ( H " C
)  /\  y  e.  ( `' S " { ( H `  D ) } ) ) )
19 r19.41v 2706 . . . . . . . . 9  |-  ( E. x  e.  C  ( ( H `  x
)  =  y  /\  y S ( H `  D ) )  <->  ( E. x  e.  C  ( H `  x )  =  y  /\  y S ( H `  D ) ) )
2017, 18, 193bitr4g 279 . . . . . . . 8  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  C  C_  A )  ->  (
y  e.  ( ( H " C )  i^i  ( `' S " { ( H `  D ) } ) )  <->  E. x  e.  C  ( ( H `  x )  =  y  /\  y S ( H `  D ) ) ) )
2120adantrr 697 . . . . . . 7  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( y  e.  ( ( H " C
)  i^i  ( `' S " { ( H `
 D ) } ) )  <->  E. x  e.  C  ( ( H `  x )  =  y  /\  y S ( H `  D ) ) ) )
22 breq1 4042 . . . . . . . . . . . . . 14  |-  ( ( H `  x )  =  y  ->  (
( H `  x
) S ( H `
 D )  <->  y S
( H `  D
) ) )
2322biimpar 471 . . . . . . . . . . . . 13  |-  ( ( ( H `  x
)  =  y  /\  y S ( H `  D ) )  -> 
( H `  x
) S ( H `
 D ) )
24 vex 2804 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
2524eliniseg 5058 . . . . . . . . . . . . . . 15  |-  ( D  e.  A  ->  (
x  e.  ( `' R " { D } )  <->  x R D ) )
2625ad2antll 709 . . . . . . . . . . . . . 14  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x  e.  ( `' R " { D } )  <->  x R D ) )
27 isorel 5839 . . . . . . . . . . . . . 14  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x R D  <->  ( H `  x ) S ( H `  D ) ) )
2826, 27bitrd 244 . . . . . . . . . . . . 13  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x  e.  ( `' R " { D } )  <->  ( H `  x ) S ( H `  D ) ) )
2923, 28syl5ibr 212 . . . . . . . . . . . 12  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( (
( H `  x
)  =  y  /\  y S ( H `  D ) )  ->  x  e.  ( `' R " { D }
) ) )
3029exp32 588 . . . . . . . . . . 11  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( x  e.  A  ->  ( D  e.  A  ->  ( ( ( H `  x
)  =  y  /\  y S ( H `  D ) )  ->  x  e.  ( `' R " { D }
) ) ) ) )
312, 30syl9r 67 . . . . . . . . . 10  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( C  C_  A  ->  ( x  e.  C  ->  ( D  e.  A  ->  ( ( ( H `  x
)  =  y  /\  y S ( H `  D ) )  ->  x  e.  ( `' R " { D }
) ) ) ) ) )
3231com34 77 . . . . . . . . 9  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( C  C_  A  ->  ( D  e.  A  ->  ( x  e.  C  ->  ( ( ( H `  x
)  =  y  /\  y S ( H `  D ) )  ->  x  e.  ( `' R " { D }
) ) ) ) ) )
3332imp32 422 . . . . . . . 8  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( x  e.  C  ->  ( ( ( H `
 x )  =  y  /\  y S ( H `  D
) )  ->  x  e.  ( `' R " { D } ) ) ) )
3433reximdvai 2666 . . . . . . 7  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( E. x  e.  C  ( ( H `
 x )  =  y  /\  y S ( H `  D
) )  ->  E. x  e.  C  x  e.  ( `' R " { D } ) ) )
3521, 34sylbid 206 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( y  e.  ( ( H " C
)  i^i  ( `' S " { ( H `
 D ) } ) )  ->  E. x  e.  C  x  e.  ( `' R " { D } ) ) )
36 elin 3371 . . . . . . . 8  |-  ( x  e.  ( C  i^i  ( `' R " { D } ) )  <->  ( x  e.  C  /\  x  e.  ( `' R " { D } ) ) )
3736exbii 1572 . . . . . . 7  |-  ( E. x  x  e.  ( C  i^i  ( `' R " { D } ) )  <->  E. x
( x  e.  C  /\  x  e.  ( `' R " { D } ) ) )
38 neq0 3478 . . . . . . 7  |-  ( -.  ( C  i^i  ( `' R " { D } ) )  =  (/) 
<->  E. x  x  e.  ( C  i^i  ( `' R " { D } ) ) )
39 df-rex 2562 . . . . . . 7  |-  ( E. x  e.  C  x  e.  ( `' R " { D } )  <->  E. x ( x  e.  C  /\  x  e.  ( `' R " { D } ) ) )
4037, 38, 393bitr4i 268 . . . . . 6  |-  ( -.  ( C  i^i  ( `' R " { D } ) )  =  (/) 
<->  E. x  e.  C  x  e.  ( `' R " { D }
) )
4135, 40syl6ibr 218 . . . . 5  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( y  e.  ( ( H " C
)  i^i  ( `' S " { ( H `
 D ) } ) )  ->  -.  ( C  i^i  ( `' R " { D } ) )  =  (/) ) )
4241exlimdv 1626 . . . 4  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( E. y  y  e.  ( ( H
" C )  i^i  ( `' S " { ( H `  D ) } ) )  ->  -.  ( C  i^i  ( `' R " { D } ) )  =  (/) ) )
431, 42syl5bi 208 . . 3  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( -.  ( ( H " C )  i^i  ( `' S " { ( H `  D ) } ) )  =  (/)  ->  -.  ( C  i^i  ( `' R " { D } ) )  =  (/) ) )
4443con4d 97 . 2  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( ( C  i^i  ( `' R " { D } ) )  =  (/)  ->  ( ( H
" C )  i^i  ( `' S " { ( H `  D ) } ) )  =  (/) ) )
453, 4syl 15 . . . . . . . . 9  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Fn  A
)
46 fnfvima 5772 . . . . . . . . . . 11  |-  ( ( H  Fn  A  /\  C  C_  A  /\  x  e.  C )  ->  ( H `  x )  e.  ( H " C
) )
47463expia 1153 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  C  C_  A )  -> 
( x  e.  C  ->  ( H `  x
)  e.  ( H
" C ) ) )
4847adantrr 697 . . . . . . . . 9  |-  ( ( H  Fn  A  /\  ( C  C_  A  /\  D  e.  A )
)  ->  ( x  e.  C  ->  ( H `
 x )  e.  ( H " C
) ) )
4945, 48sylan 457 . . . . . . . 8  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( x  e.  C  ->  ( H `  x
)  e.  ( H
" C ) ) )
5049adantrd 454 . . . . . . 7  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( ( x  e.  C  /\  x  e.  ( `' R " { D } ) )  ->  ( H `  x )  e.  ( H " C ) ) )
5127biimpd 198 . . . . . . . . . . . . . 14  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x R D  ->  ( H `
 x ) S ( H `  D
) ) )
52 fvex 5555 . . . . . . . . . . . . . . . 16  |-  ( H `
 x )  e. 
_V
5352eliniseg 5058 . . . . . . . . . . . . . . 15  |-  ( ( H `  D )  e.  _V  ->  (
( H `  x
)  e.  ( `' S " { ( H `  D ) } )  <->  ( H `  x ) S ( H `  D ) ) )
5414, 53ax-mp 8 . . . . . . . . . . . . . 14  |-  ( ( H `  x )  e.  ( `' S " { ( H `  D ) } )  <-> 
( H `  x
) S ( H `
 D ) )
5551, 54syl6ibr 218 . . . . . . . . . . . . 13  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x R D  ->  ( H `
 x )  e.  ( `' S " { ( H `  D ) } ) ) )
5626, 55sylbid 206 . . . . . . . . . . . 12  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x  e.  ( `' R " { D } )  -> 
( H `  x
)  e.  ( `' S " { ( H `  D ) } ) ) )
5756exp32 588 . . . . . . . . . . 11  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( x  e.  A  ->  ( D  e.  A  ->  ( x  e.  ( `' R " { D } )  ->  ( H `  x )  e.  ( `' S " { ( H `  D ) } ) ) ) ) )
582, 57syl9r 67 . . . . . . . . . 10  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( C  C_  A  ->  ( x  e.  C  ->  ( D  e.  A  ->  ( x  e.  ( `' R " { D } )  ->  ( H `  x )  e.  ( `' S " { ( H `  D ) } ) ) ) ) ) )
5958com34 77 . . . . . . . . 9  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( C  C_  A  ->  ( D  e.  A  ->  ( x  e.  C  ->  ( x  e.  ( `' R " { D } )  ->  ( H `  x )  e.  ( `' S " { ( H `  D ) } ) ) ) ) ) )
6059imp32 422 . . . . . . . 8  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( x  e.  C  ->  ( x  e.  ( `' R " { D } )  ->  ( H `  x )  e.  ( `' S " { ( H `  D ) } ) ) ) )
6160imp3a 420 . . . . . . 7  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( ( x  e.  C  /\  x  e.  ( `' R " { D } ) )  ->  ( H `  x )  e.  ( `' S " { ( H `  D ) } ) ) )
6250, 61jcad 519 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( ( x  e.  C  /\  x  e.  ( `' R " { D } ) )  ->  ( ( H `
 x )  e.  ( H " C
)  /\  ( H `  x )  e.  ( `' S " { ( H `  D ) } ) ) ) )
63 elin 3371 . . . . . 6  |-  ( ( H `  x )  e.  ( ( H
" C )  i^i  ( `' S " { ( H `  D ) } ) )  <->  ( ( H `
 x )  e.  ( H " C
)  /\  ( H `  x )  e.  ( `' S " { ( H `  D ) } ) ) )
6462, 36, 633imtr4g 261 . . . . 5  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( x  e.  ( C  i^i  ( `' R " { D } ) )  -> 
( H `  x
)  e.  ( ( H " C )  i^i  ( `' S " { ( H `  D ) } ) ) ) )
65 n0i 3473 . . . . 5  |-  ( ( H `  x )  e.  ( ( H
" C )  i^i  ( `' S " { ( H `  D ) } ) )  ->  -.  (
( H " C
)  i^i  ( `' S " { ( H `
 D ) } ) )  =  (/) )
6664, 65syl6 29 . . . 4  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( x  e.  ( C  i^i  ( `' R " { D } ) )  ->  -.  ( ( H " C )  i^i  ( `' S " { ( H `  D ) } ) )  =  (/) ) )
6766exlimdv 1626 . . 3  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( E. x  x  e.  ( C  i^i  ( `' R " { D } ) )  ->  -.  ( ( H " C )  i^i  ( `' S " { ( H `  D ) } ) )  =  (/) ) )
6838, 67syl5bi 208 . 2  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( -.  ( C  i^i  ( `' R " { D } ) )  =  (/)  ->  -.  ( ( H " C )  i^i  ( `' S " { ( H `  D ) } ) )  =  (/) ) )
6944, 68impcon4bid 196 1  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( ( C  i^i  ( `' R " { D } ) )  =  (/) 
<->  ( ( H " C )  i^i  ( `' S " { ( H `  D ) } ) )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   class class class wbr 4039   `'ccnv 4704   "cima 4708    Fn wfn 5266   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272
This theorem is referenced by:  isofrlem  5853
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-f1o 5278  df-fv 5279  df-isom 5280
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