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Theorem isocnv2 5807
Description: Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
Assertion
Ref Expression
isocnv2  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  `' R ,  `' S ( A ,  B ) )

Proof of Theorem isocnv2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isof1o 5802 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
2 f1ofn 5458 . . 3  |-  ( H : A -1-1-onto-> B  ->  H  Fn  A )
31, 2syl 14 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H  Fn  A
)
4 isof1o 5802 . . 3  |-  ( H 
Isom  `' R ,  `' S
( A ,  B
)  ->  H : A
-1-1-onto-> B )
54, 2syl 14 . 2  |-  ( H 
Isom  `' R ,  `' S
( A ,  B
)  ->  H  Fn  A )
6 ralcom 2640 . . . . 5  |-  ( A. y  e.  A  A. x  e.  A  (
y R x  <->  ( H `  y ) S ( H `  x ) )  <->  A. x  e.  A  A. y  e.  A  ( y R x  <-> 
( H `  y
) S ( H `
 x ) ) )
7 vex 2740 . . . . . . . . . 10  |-  x  e. 
_V
8 vex 2740 . . . . . . . . . 10  |-  y  e. 
_V
97, 8brcnv 4806 . . . . . . . . 9  |-  ( x `' R y  <->  y R x )
109a1i 9 . . . . . . . 8  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  (
x `' R y  <-> 
y R x ) )
11 funfvex 5528 . . . . . . . . . . 11  |-  ( ( Fun  H  /\  x  e.  dom  H )  -> 
( H `  x
)  e.  _V )
1211funfni 5312 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  x  e.  A )  ->  ( H `  x
)  e.  _V )
1312adantr 276 . . . . . . . . 9  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  ( H `  x )  e.  _V )
14 funfvex 5528 . . . . . . . . . . 11  |-  ( ( Fun  H  /\  y  e.  dom  H )  -> 
( H `  y
)  e.  _V )
1514funfni 5312 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  y  e.  A )  ->  ( H `  y
)  e.  _V )
1615adantlr 477 . . . . . . . . 9  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  ( H `  y )  e.  _V )
17 brcnvg 4804 . . . . . . . . 9  |-  ( ( ( H `  x
)  e.  _V  /\  ( H `  y )  e.  _V )  -> 
( ( H `  x ) `' S
( H `  y
)  <->  ( H `  y ) S ( H `  x ) ) )
1813, 16, 17syl2anc 411 . . . . . . . 8  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  (
( H `  x
) `' S ( H `  y )  <-> 
( H `  y
) S ( H `
 x ) ) )
1910, 18bibi12d 235 . . . . . . 7  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  (
( x `' R
y  <->  ( H `  x ) `' S
( H `  y
) )  <->  ( y R x  <->  ( H `  y ) S ( H `  x ) ) ) )
2019ralbidva 2473 . . . . . 6  |-  ( ( H  Fn  A  /\  x  e.  A )  ->  ( A. y  e.  A  ( x `' R y  <->  ( H `  x ) `' S
( H `  y
) )  <->  A. y  e.  A  ( y R x  <->  ( H `  y ) S ( H `  x ) ) ) )
2120ralbidva 2473 . . . . 5  |-  ( H  Fn  A  ->  ( A. x  e.  A  A. y  e.  A  ( x `' R
y  <->  ( H `  x ) `' S
( H `  y
) )  <->  A. x  e.  A  A. y  e.  A  ( y R x  <->  ( H `  y ) S ( H `  x ) ) ) )
226, 21bitr4id 199 . . . 4  |-  ( H  Fn  A  ->  ( A. y  e.  A  A. x  e.  A  ( y R x  <-> 
( H `  y
) S ( H `
 x ) )  <->  A. x  e.  A  A. y  e.  A  ( x `' R
y  <->  ( H `  x ) `' S
( H `  y
) ) ) )
2322anbi2d 464 . . 3  |-  ( H  Fn  A  ->  (
( H : A -1-1-onto-> B  /\  A. y  e.  A  A. x  e.  A  ( y R x  <-> 
( H `  y
) S ( H `
 x ) ) )  <->  ( H : A
-1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x `' R y  <->  ( H `  x ) `' S
( H `  y
) ) ) ) )
24 df-isom 5221 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
( H : A -1-1-onto-> B  /\  A. y  e.  A  A. x  e.  A  ( y R x  <-> 
( H `  y
) S ( H `
 x ) ) ) )
25 df-isom 5221 . . 3  |-  ( H 
Isom  `' R ,  `' S
( A ,  B
)  <->  ( H : A
-1-1-onto-> B  /\  A. x  e.  A  A. y  e.  A  ( x `' R y  <->  ( H `  x ) `' S
( H `  y
) ) ) )
2623, 24, 253bitr4g 223 . 2  |-  ( H  Fn  A  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  `' R ,  `' S ( A ,  B ) ) )
273, 5, 26pm5.21nii 704 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  `' R ,  `' S ( A ,  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    e. wcel 2148   A.wral 2455   _Vcvv 2737   class class class wbr 4000   `'ccnv 4622    Fn wfn 5207   -1-1-onto->wf1o 5211   ` cfv 5212    Isom wiso 5213
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-sep 4118  ax-pow 4171  ax-pr 4206
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-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-f1o 5219  df-fv 5220  df-isom 5221
This theorem is referenced by:  infisoti  7025
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