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Theorem isofr 5839
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 5827 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
2 id 19 . . . 4  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
3 isof1o 5822 . . . . 5  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  `' H : B -1-1-onto-> A )
4 f1ofun 5474 . . . . 5  |-  ( `' H : B -1-1-onto-> A  ->  Fun  `' H )
5 vex 2791 . . . . . 6  |-  x  e. 
_V
65funimaex 5330 . . . . 5  |-  ( Fun  `' H  ->  ( `' H " x )  e.  _V )
73, 4, 63syl 18 . . . 4  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  ( `' H " x )  e.  _V )
82, 7isofrlem 5837 . . 3  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  ( R  Fr  A  ->  S  Fr  B ) )
91, 8syl 15 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Fr  A  ->  S  Fr  B
) )
10 id 19 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Isom  R ,  S  ( A ,  B ) )
11 isof1o 5822 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
12 f1ofun 5474 . . . 4  |-  ( H : A -1-1-onto-> B  ->  Fun  H )
135funimaex 5330 . . . 4  |-  ( Fun 
H  ->  ( H " x )  e.  _V )
1411, 12, 133syl 18 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( H "
x )  e.  _V )
1510, 14isofrlem 5837 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Fr  B  ->  R  Fr  A
) )
169, 15impbid 183 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Fr  A 
<->  S  Fr  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1684   _Vcvv 2788    Fr wfr 4349   `'ccnv 4688   "cima 4692   Fun wfun 5249   -1-1-onto->wf1o 5254    Isom wiso 5256
This theorem is referenced by:  isowe  5846  wofib  7260  isfin1-4  8013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-fr 4352  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264
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