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Theorem isofr 6213
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 6201 . . 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 6196 . . . . 5  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  `' H : B -1-1-onto-> A )
4 f1ofun 5800 . . . . 5  |-  ( `' H : B -1-1-onto-> A  ->  Fun  `' H )
5 vex 3109 . . . . . 6  |-  x  e. 
_V
65funimaex 5648 . . . . 5  |-  ( Fun  `' H  ->  ( `' H " x )  e.  _V )
73, 4, 63syl 20 . . . 4  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  ( `' H " x )  e.  _V )
82, 7isofrlem 6211 . . 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 6196 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
12 f1ofun 5800 . . . 4  |-  ( H : A -1-1-onto-> B  ->  Fun  H )
135funimaex 5648 . . . 4  |-  ( Fun 
H  ->  ( H " x )  e.  _V )
1411, 12, 133syl 20 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( H "
x )  e.  _V )
1510, 14isofrlem 6211 . 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 1823   _Vcvv 3106    Fr wfr 4824   `'ccnv 4987   "cima 4991   Fun wfun 5564   -1-1-onto->wf1o 5569    Isom wiso 5571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-fr 4827  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579
This theorem is referenced by:  isowe  6220  wofib  7962  isfin1-4  8758
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