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Theorem isose 5882
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 5864 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
3 f1ofun 5512 . . . 4  |-  ( H : A -1-1-onto-> B  ->  Fun  H )
4 vex 2825 . . . . 5  |-  x  e. 
_V
54funimaex 5367 . . . 4  |-  ( Fun 
H  ->  ( H " x )  e.  _V )
62, 3, 53syl 18 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( H "
x )  e.  _V )
71, 6isoselem 5880 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R Se  A  ->  S Se  B ) )
8 isocnv 5869 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
9 isof1o 5864 . . . . 5  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  `' H : B -1-1-onto-> A )
10 f1ofun 5512 . . . . 5  |-  ( `' H : B -1-1-onto-> A  ->  Fun  `' H )
114funimaex 5367 . . . . 5  |-  ( Fun  `' H  ->  ( `' H " x )  e.  _V )
129, 10, 113syl 18 . . . 4  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  ( `' H " x )  e.  _V )
138, 12syl 15 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( `' H " x )  e.  _V )
148, 13isoselem 5880 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S Se  B  ->  R Se  A ) )
157, 14impbid 183 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R Se  A  <->  S Se  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1701   _Vcvv 2822   Se wse 4387   `'ccnv 4725   "cima 4729   Fun wfun 5286   -1-1-onto->wf1o 5291    Isom wiso 5293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-se 4390  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301
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