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Theorem isofr 6137
Description: An isomorphism preserves well-foundedness. Proposition 6.32(1) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
Assertion
Ref Expression
isofr  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Fr  A 
<->  S  Fr  B ) )

Proof of Theorem isofr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 isocnv 6125 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
2 id 22 . . . 4  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
3 isof1o 6120 . . . . 5  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  `' H : B -1-1-onto-> A )
4 f1ofun 5746 . . . . 5  |-  ( `' H : B -1-1-onto-> A  ->  Fun  `' H )
5 vex 3075 . . . . . 6  |-  x  e. 
_V
65funimaex 5599 . . . . 5  |-  ( Fun  `' H  ->  ( `' H " x )  e.  _V )
73, 4, 63syl 20 . . . 4  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  ( `' H " x )  e.  _V )
82, 7isofrlem 6135 . . 3  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  ( R  Fr  A  ->  S  Fr  B ) )
91, 8syl 16 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Fr  A  ->  S  Fr  B
) )
10 id 22 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Isom  R ,  S  ( A ,  B ) )
11 isof1o 6120 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
12 f1ofun 5746 . . . 4  |-  ( H : A -1-1-onto-> B  ->  Fun  H )
135funimaex 5599 . . . 4  |-  ( Fun 
H  ->  ( H " x )  e.  _V )
1411, 12, 133syl 20 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( H "
x )  e.  _V )
1510, 14isofrlem 6135 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Fr  B  ->  R  Fr  A
) )
169, 15impbid 191 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Fr  A 
<->  S  Fr  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1758   _Vcvv 3072    Fr wfr 4779   `'ccnv 4942   "cima 4946   Fun wfun 5515   -1-1-onto->wf1o 5520    Isom wiso 5522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-id 4739  df-fr 4782  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-isom 5530
This theorem is referenced by:  isowe  6144  wofib  7865  isfin1-4  8662
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