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Theorem isofr 5801
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 ) )
Dummy variable  x is distinct from all other variables.

Proof of Theorem isofr
StepHypRef Expression
1 isocnv 5789 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
2 id 21 . . . 4  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
3 isof1o 5784 . . . . 5  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  `' H : B -1-1-onto-> A )
4 f1ofun 5440 . . . . 5  |-  ( `' H : B -1-1-onto-> A  ->  Fun  `' H )
5 vex 2793 . . . . . 6  |-  x  e. 
_V
65funimaex 5296 . . . . 5  |-  ( Fun  `' H  ->  ( `' H " x )  e.  _V )
73, 4, 63syl 20 . . . 4  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  ( `' H " x )  e.  _V )
82, 7isofrlem 5799 . . 3  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  ( R  Fr  A  ->  S  Fr  B ) )
91, 8syl 17 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Fr  A  ->  S  Fr  B
) )
10 id 21 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Isom  R ,  S  ( A ,  B ) )
11 isof1o 5784 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
12 f1ofun 5440 . . . 4  |-  ( H : A -1-1-onto-> B  ->  Fun  H )
135funimaex 5296 . . . 4  |-  ( Fun 
H  ->  ( H " x )  e.  _V )
1411, 12, 133syl 20 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( H "
x )  e.  _V )
1510, 14isofrlem 5799 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Fr  B  ->  R  Fr  A
) )
169, 15impbid 185 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R  Fr  A 
<->  S  Fr  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    e. wcel 1685   _Vcvv 2790    Fr wfr 4349   `'ccnv 4688   "cima 4692   Fun wfun 5216   -1-1-onto->wf1o 5221    Isom wiso 5223
This theorem is referenced by:  isowe  5808  wofib  7256  isfin1-4  8009
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231
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