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Theorem isomin 6060
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 3640 . . . 4  |-  ( -.  ( ( H " C )  i^i  ( `' S " { ( H `  D ) } ) )  =  (/) 
<->  E. y  y  e.  ( ( H " C )  i^i  ( `' S " { ( H `  D ) } ) ) )
2 ssel 3344 . . . . . . . . . . . . . 14  |-  ( C 
C_  A  ->  (
x  e.  C  ->  x  e.  A )
)
3 isof1o 6048 . . . . . . . . . . . . . . 15  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
4 f1ofn 5678 . . . . . . . . . . . . . . 15  |-  ( H : A -1-1-onto-> B  ->  H  Fn  A )
5 fnbrfvb 5770 . . . . . . . . . . . . . . . 16  |-  ( ( H  Fn  A  /\  x  e.  A )  ->  ( ( H `  x )  =  y  <-> 
x H y ) )
65ex 425 . . . . . . . . . . . . . . 15  |-  ( H  Fn  A  ->  (
x  e.  A  -> 
( ( H `  x )  =  y  <-> 
x H y ) ) )
73, 4, 63syl 19 . . . . . . . . . . . . . 14  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( x  e.  A  ->  ( ( H `  x )  =  y  <->  x H y ) ) )
82, 7syl9r 70 . . . . . . . . . . . . 13  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( C  C_  A  ->  ( x  e.  C  ->  ( ( H `  x )  =  y  <->  x H y ) ) ) )
98imp31 423 . . . . . . . . . . . 12  |-  ( ( ( H  Isom  R ,  S  ( A ,  B )  /\  C  C_  A )  /\  x  e.  C )  ->  (
( H `  x
)  =  y  <->  x H
y ) )
109rexbidva 2724 . . . . . . . . . . 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 2961 . . . . . . . . . . . 12  |-  y  e. 
_V
1211elima 5211 . . . . . . . . . . 11  |-  ( y  e.  ( H " C )  <->  E. x  e.  C  x H
y )
1310, 12syl6rbbr 257 . . . . . . . . . 10  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  C  C_  A )  ->  (
y  e.  ( H
" C )  <->  E. x  e.  C  ( H `  x )  =  y ) )
14 fvex 5745 . . . . . . . . . . 11  |-  ( H `
 D )  e. 
_V
1511eliniseg 5236 . . . . . . . . . . 11  |-  ( ( H `  D )  e.  _V  ->  (
y  e.  ( `' S " { ( H `  D ) } )  <->  y S
( H `  D
) ) )
1614, 15mp1i 12 . . . . . . . . . 10  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  C  C_  A )  ->  (
y  e.  ( `' S " { ( H `  D ) } )  <->  y S
( H `  D
) ) )
1713, 16anbi12d 693 . . . . . . . . 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 3532 . . . . . . . . 9  |-  ( y  e.  ( ( H
" C )  i^i  ( `' S " { ( H `  D ) } ) )  <->  ( y  e.  ( H " C
)  /\  y  e.  ( `' S " { ( H `  D ) } ) ) )
19 r19.41v 2863 . . . . . . . . 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 281 . . . . . . . 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 699 . . . . . . 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 4218 . . . . . . . . . . . . . 14  |-  ( ( H `  x )  =  y  ->  (
( H `  x
) S ( H `
 D )  <->  y S
( H `  D
) ) )
2322biimpar 473 . . . . . . . . . . . . 13  |-  ( ( ( H `  x
)  =  y  /\  y S ( H `  D ) )  -> 
( H `  x
) S ( H `
 D ) )
24 vex 2961 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
2524eliniseg 5236 . . . . . . . . . . . . . . 15  |-  ( D  e.  A  ->  (
x  e.  ( `' R " { D } )  <->  x R D ) )
2625ad2antll 711 . . . . . . . . . . . . . 14  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x  e.  ( `' R " { D } )  <->  x R D ) )
27 isorel 6049 . . . . . . . . . . . . . 14  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x R D  <->  ( H `  x ) S ( H `  D ) ) )
2826, 27bitrd 246 . . . . . . . . . . . . 13  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x  e.  ( `' R " { D } )  <->  ( H `  x ) S ( H `  D ) ) )
2923, 28syl5ibr 214 . . . . . . . . . . . 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 590 . . . . . . . . . . 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 70 . . . . . . . . . 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 80 . . . . . . . . 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 424 . . . . . . . 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 2818 . . . . . . 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 208 . . . . . 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 3532 . . . . . . . 8  |-  ( x  e.  ( C  i^i  ( `' R " { D } ) )  <->  ( x  e.  C  /\  x  e.  ( `' R " { D } ) ) )
3736exbii 1593 . . . . . . 7  |-  ( E. x  x  e.  ( C  i^i  ( `' R " { D } ) )  <->  E. x
( x  e.  C  /\  x  e.  ( `' R " { D } ) ) )
38 neq0 3640 . . . . . . 7  |-  ( -.  ( C  i^i  ( `' R " { D } ) )  =  (/) 
<->  E. x  x  e.  ( C  i^i  ( `' R " { D } ) ) )
39 df-rex 2713 . . . . . . 7  |-  ( E. x  e.  C  x  e.  ( `' R " { D } )  <->  E. x ( x  e.  C  /\  x  e.  ( `' R " { D } ) ) )
4037, 38, 393bitr4i 270 . . . . . 6  |-  ( -.  ( C  i^i  ( `' R " { D } ) )  =  (/) 
<->  E. x  e.  C  x  e.  ( `' R " { D }
) )
4135, 40syl6ibr 220 . . . . 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 1647 . . . 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 210 . . 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 100 . 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 16 . . . . . . . . 9  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Fn  A
)
46 fnfvima 5979 . . . . . . . . . . 11  |-  ( ( H  Fn  A  /\  C  C_  A  /\  x  e.  C )  ->  ( H `  x )  e.  ( H " C
) )
47463expia 1156 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  C  C_  A )  -> 
( x  e.  C  ->  ( H `  x
)  e.  ( H
" C ) ) )
4847adantrr 699 . . . . . . . . 9  |-  ( ( H  Fn  A  /\  ( C  C_  A  /\  D  e.  A )
)  ->  ( x  e.  C  ->  ( H `
 x )  e.  ( H " C
) ) )
4945, 48sylan 459 . . . . . . . 8  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( x  e.  C  ->  ( H `  x
)  e.  ( H
" C ) ) )
5049adantrd 456 . . . . . . 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 200 . . . . . . . . . . . . . 14  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x R D  ->  ( H `
 x ) S ( H `  D
) ) )
52 fvex 5745 . . . . . . . . . . . . . . . 16  |-  ( H `
 x )  e. 
_V
5352eliniseg 5236 . . . . . . . . . . . . . . 15  |-  ( ( H `  D )  e.  _V  ->  (
( H `  x
)  e.  ( `' S " { ( H `  D ) } )  <->  ( H `  x ) S ( H `  D ) ) )
5414, 53ax-mp 5 . . . . . . . . . . . . . 14  |-  ( ( H `  x )  e.  ( `' S " { ( H `  D ) } )  <-> 
( H `  x
) S ( H `
 D ) )
5551, 54syl6ibr 220 . . . . . . . . . . . . 13  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x R D  ->  ( H `
 x )  e.  ( `' S " { ( H `  D ) } ) ) )
5626, 55sylbid 208 . . . . . . . . . . . 12  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x  e.  ( `' R " { D } )  -> 
( H `  x
)  e.  ( `' S " { ( H `  D ) } ) ) )
5756exp32 590 . . . . . . . . . . 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 70 . . . . . . . . . 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 80 . . . . . . . . 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 424 . . . . . . . 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 422 . . . . . . 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 521 . . . . . 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 3532 . . . . . 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 263 . . . . 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 3635 . . . . 5  |-  ( ( H `  x )  e.  ( ( H
" C )  i^i  ( `' S " { ( H `  D ) } ) )  ->  -.  (
( H " C
)  i^i  ( `' S " { ( H `
 D ) } ) )  =  (/) )
6664, 65syl6 32 . . . 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 1647 . . 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 210 . 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 198 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 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   E.wrex 2708   _Vcvv 2958    i^i cin 3321    C_ wss 3322   (/)c0 3630   {csn 3816   class class class wbr 4215   `'ccnv 4880   "cima 4884    Fn wfn 5452   -1-1-onto->wf1o 5456   ` cfv 5457    Isom wiso 5458
This theorem is referenced by:  isofrlem  6063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-f1o 5464  df-fv 5465  df-isom 5466
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