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Theorem isose 5821
Description: An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
Assertion
Ref Expression
isose  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R Se  A  <->  S Se  B ) )

Proof of Theorem isose
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 19 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Isom  R ,  S  ( A ,  B ) )
2 isof1o 5807 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
3 f1ofun 5463 . . . 4  |-  ( H : A -1-1-onto-> B  ->  Fun  H )
4 vex 2740 . . . . 5  |-  x  e. 
_V
54funimaex 5301 . . . 4  |-  ( Fun 
H  ->  ( H " x )  e.  _V )
62, 3, 53syl 17 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( H "
x )  e.  _V )
71, 6isoselem 5820 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R Se  A  ->  S Se  B ) )
8 isocnv 5811 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
9 isof1o 5807 . . . 4  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  `' H : B -1-1-onto-> A )
10 f1ofun 5463 . . . 4  |-  ( `' H : B -1-1-onto-> A  ->  Fun  `' H )
114funimaex 5301 . . . 4  |-  ( Fun  `' H  ->  ( `' H " x )  e.  _V )
128, 9, 10, 114syl 18 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( `' H " x )  e.  _V )
138, 12isoselem 5820 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S Se  B  ->  R Se  A ) )
147, 13impbid 129 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R Se  A  <->  S Se  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    e. wcel 2148   _Vcvv 2737   Se wse 4329   `'ccnv 4625   "cima 4629   Fun wfun 5210   -1-1-onto->wf1o 5215    Isom wiso 5217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-se 4333  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225
This theorem is referenced by: (None)
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