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Theorem isofr 5855
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 5843 . . 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 5838 . . . . 5  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  `' H : B -1-1-onto-> A )
4 f1ofun 5490 . . . . 5  |-  ( `' H : B -1-1-onto-> A  ->  Fun  `' H )
5 vex 2804 . . . . . 6  |-  x  e. 
_V
65funimaex 5346 . . . . 5  |-  ( Fun  `' H  ->  ( `' H " x )  e.  _V )
73, 4, 63syl 18 . . . 4  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  ( `' H " x )  e.  _V )
82, 7isofrlem 5853 . . 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 5838 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
12 f1ofun 5490 . . . 4  |-  ( H : A -1-1-onto-> B  ->  Fun  H )
135funimaex 5346 . . . 4  |-  ( Fun 
H  ->  ( H " x )  e.  _V )
1411, 12, 133syl 18 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( H "
x )  e.  _V )
1510, 14isofrlem 5853 . 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 1696   _Vcvv 2801    Fr wfr 4365   `'ccnv 4704   "cima 4708   Fun wfun 5265   -1-1-onto->wf1o 5270    Isom wiso 5272
This theorem is referenced by:  isowe  5862  wofib  7276  isfin1-4  8029
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-fr 4368  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280
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