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Theorem isomin 6048
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 3630 . . . 4  |-  ( -.  ( ( H " C )  i^i  ( `' S " { ( H `  D ) } ) )  =  (/) 
<->  E. y  y  e.  ( ( H " C )  i^i  ( `' S " { ( H `  D ) } ) ) )
2 ssel 3334 . . . . . . . . . . . . . 14  |-  ( C 
C_  A  ->  (
x  e.  C  ->  x  e.  A )
)
3 isof1o 6036 . . . . . . . . . . . . . . 15  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
4 f1ofn 5666 . . . . . . . . . . . . . . 15  |-  ( H : A -1-1-onto-> B  ->  H  Fn  A )
5 fnbrfvb 5758 . . . . . . . . . . . . . . . 16  |-  ( ( H  Fn  A  /\  x  e.  A )  ->  ( ( H `  x )  =  y  <-> 
x H y ) )
65ex 424 . . . . . . . . . . . . . . 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 69 . . . . . . . . . . . . 13  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( C  C_  A  ->  ( x  e.  C  ->  ( ( H `  x )  =  y  <->  x H y ) ) ) )
98imp31 422 . . . . . . . . . . . 12  |-  ( ( ( H  Isom  R ,  S  ( A ,  B )  /\  C  C_  A )  /\  x  e.  C )  ->  (
( H `  x
)  =  y  <->  x H
y ) )
109rexbidva 2714 . . . . . . . . . . 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 2951 . . . . . . . . . . . 12  |-  y  e. 
_V
1211elima 5199 . . . . . . . . . . 11  |-  ( y  e.  ( H " C )  <->  E. x  e.  C  x H
y )
1310, 12syl6rbbr 256 . . . . . . . . . 10  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  C  C_  A )  ->  (
y  e.  ( H
" C )  <->  E. x  e.  C  ( H `  x )  =  y ) )
14 fvex 5733 . . . . . . . . . . 11  |-  ( H `
 D )  e. 
_V
1511eliniseg 5224 . . . . . . . . . . 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 692 . . . . . . . . 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 3522 . . . . . . . . 9  |-  ( y  e.  ( ( H
" C )  i^i  ( `' S " { ( H `  D ) } ) )  <->  ( y  e.  ( H " C
)  /\  y  e.  ( `' S " { ( H `  D ) } ) ) )
19 r19.41v 2853 . . . . . . . . 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 280 . . . . . . . 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 698 . . . . . . 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 4207 . . . . . . . . . . . . . 14  |-  ( ( H `  x )  =  y  ->  (
( H `  x
) S ( H `
 D )  <->  y S
( H `  D
) ) )
2322biimpar 472 . . . . . . . . . . . . 13  |-  ( ( ( H `  x
)  =  y  /\  y S ( H `  D ) )  -> 
( H `  x
) S ( H `
 D ) )
24 vex 2951 . . . . . . . . . . . . . . . 16  |-  x  e. 
_V
2524eliniseg 5224 . . . . . . . . . . . . . . 15  |-  ( D  e.  A  ->  (
x  e.  ( `' R " { D } )  <->  x R D ) )
2625ad2antll 710 . . . . . . . . . . . . . 14  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x  e.  ( `' R " { D } )  <->  x R D ) )
27 isorel 6037 . . . . . . . . . . . . . 14  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x R D  <->  ( H `  x ) S ( H `  D ) ) )
2826, 27bitrd 245 . . . . . . . . . . . . 13  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x  e.  ( `' R " { D } )  <->  ( H `  x ) S ( H `  D ) ) )
2923, 28syl5ibr 213 . . . . . . . . . . . 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 589 . . . . . . . . . . 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 69 . . . . . . . . . 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 79 . . . . . . . . 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 423 . . . . . . . 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 2808 . . . . . . 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 207 . . . . . 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 3522 . . . . . . . 8  |-  ( x  e.  ( C  i^i  ( `' R " { D } ) )  <->  ( x  e.  C  /\  x  e.  ( `' R " { D } ) ) )
3736exbii 1592 . . . . . . 7  |-  ( E. x  x  e.  ( C  i^i  ( `' R " { D } ) )  <->  E. x
( x  e.  C  /\  x  e.  ( `' R " { D } ) ) )
38 neq0 3630 . . . . . . 7  |-  ( -.  ( C  i^i  ( `' R " { D } ) )  =  (/) 
<->  E. x  x  e.  ( C  i^i  ( `' R " { D } ) ) )
39 df-rex 2703 . . . . . . 7  |-  ( E. x  e.  C  x  e.  ( `' R " { D } )  <->  E. x ( x  e.  C  /\  x  e.  ( `' R " { D } ) ) )
4037, 38, 393bitr4i 269 . . . . . 6  |-  ( -.  ( C  i^i  ( `' R " { D } ) )  =  (/) 
<->  E. x  e.  C  x  e.  ( `' R " { D }
) )
4135, 40syl6ibr 219 . . . . 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 1646 . . . 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 209 . . 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 99 . 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 5967 . . . . . . . . . . 11  |-  ( ( H  Fn  A  /\  C  C_  A  /\  x  e.  C )  ->  ( H `  x )  e.  ( H " C
) )
47463expia 1155 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  C  C_  A )  -> 
( x  e.  C  ->  ( H `  x
)  e.  ( H
" C ) ) )
4847adantrr 698 . . . . . . . . 9  |-  ( ( H  Fn  A  /\  ( C  C_  A  /\  D  e.  A )
)  ->  ( x  e.  C  ->  ( H `
 x )  e.  ( H " C
) ) )
4945, 48sylan 458 . . . . . . . 8  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  ( C  C_  A  /\  D  e.  A ) )  -> 
( x  e.  C  ->  ( H `  x
)  e.  ( H
" C ) ) )
5049adantrd 455 . . . . . . 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 199 . . . . . . . . . . . . . 14  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x R D  ->  ( H `
 x ) S ( H `  D
) ) )
52 fvex 5733 . . . . . . . . . . . . . . . 16  |-  ( H `
 x )  e. 
_V
5352eliniseg 5224 . . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . 13  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x R D  ->  ( H `
 x )  e.  ( `' S " { ( H `  D ) } ) ) )
5626, 55sylbid 207 . . . . . . . . . . . 12  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
x  e.  A  /\  D  e.  A )
)  ->  ( x  e.  ( `' R " { D } )  -> 
( H `  x
)  e.  ( `' S " { ( H `  D ) } ) ) )
5756exp32 589 . . . . . . . . . . 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 69 . . . . . . . . . 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 79 . . . . . . . . 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 423 . . . . . . . 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 421 . . . . . . 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 520 . . . . . 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 3522 . . . . . 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 262 . . . . 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 3625 . . . . 5  |-  ( ( H `  x )  e.  ( ( H
" C )  i^i  ( `' S " { ( H `  D ) } ) )  ->  -.  (
( H " C
)  i^i  ( `' S " { ( H `
 D ) } ) )  =  (/) )
6664, 65syl6 31 . . . 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 1646 . . 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 209 . 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 197 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 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204   `'ccnv 4868   "cima 4872    Fn wfn 5440   -1-1-onto->wf1o 5444   ` cfv 5445    Isom wiso 5446
This theorem is referenced by:  isofrlem  6051
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-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-f1o 5452  df-fv 5453  df-isom 5454
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