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Theorem isose 6064
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 21 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Isom  R ,  S  ( A ,  B ) )
2 isof1o 6046 . . . 4  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
3 f1ofun 5677 . . . 4  |-  ( H : A -1-1-onto-> B  ->  Fun  H )
4 vex 2960 . . . . 5  |-  x  e. 
_V
54funimaex 5532 . . . 4  |-  ( Fun 
H  ->  ( H " x )  e.  _V )
62, 3, 53syl 19 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( H "
x )  e.  _V )
71, 6isoselem 6062 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R Se  A  ->  S Se  B ) )
8 isocnv 6051 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  `' H  Isom  S ,  R  ( B ,  A ) )
9 isof1o 6046 . . . 4  |-  ( `' H  Isom  S ,  R  ( B ,  A )  ->  `' H : B -1-1-onto-> A )
10 f1ofun 5677 . . . 4  |-  ( `' H : B -1-1-onto-> A  ->  Fun  `' H )
114funimaex 5532 . . . 4  |-  ( Fun  `' H  ->  ( `' H " x )  e.  _V )
128, 9, 10, 114syl 20 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( `' H " x )  e.  _V )
138, 12isoselem 6062 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S Se  B  ->  R Se  A ) )
147, 13impbid 185 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( R Se  A  <->  S Se  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    e. wcel 1726   _Vcvv 2957   Se wse 4540   `'ccnv 4878   "cima 4882   Fun wfun 5449   -1-1-onto->wf1o 5454    Isom wiso 5456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-se 4543  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464
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