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Theorem isocnv2 5591
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 5586 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
2 f1ofn 5254 . . 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 5586 . . 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 vex 2622 . . . . . . . . . 10  |-  x  e. 
_V
7 vex 2622 . . . . . . . . . 10  |-  y  e. 
_V
86, 7brcnv 4619 . . . . . . . . 9  |-  ( x `' R y  <->  y R x )
98a1i 9 . . . . . . . 8  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  (
x `' R y  <-> 
y R x ) )
10 funfvex 5322 . . . . . . . . . . 11  |-  ( ( Fun  H  /\  x  e.  dom  H )  -> 
( H `  x
)  e.  _V )
1110funfni 5114 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  x  e.  A )  ->  ( H `  x
)  e.  _V )
1211adantr 270 . . . . . . . . 9  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  ( H `  x )  e.  _V )
13 funfvex 5322 . . . . . . . . . . 11  |-  ( ( Fun  H  /\  y  e.  dom  H )  -> 
( H `  y
)  e.  _V )
1413funfni 5114 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  y  e.  A )  ->  ( H `  y
)  e.  _V )
1514adantlr 461 . . . . . . . . 9  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  ( H `  y )  e.  _V )
16 brcnvg 4617 . . . . . . . . 9  |-  ( ( ( H `  x
)  e.  _V  /\  ( H `  y )  e.  _V )  -> 
( ( H `  x ) `' S
( H `  y
)  <->  ( H `  y ) S ( H `  x ) ) )
1712, 15, 16syl2anc 403 . . . . . . . 8  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  (
( H `  x
) `' S ( H `  y )  <-> 
( H `  y
) S ( H `
 x ) ) )
189, 17bibi12d 233 . . . . . . 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 ) ) ) )
1918ralbidva 2376 . . . . . 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 ) ) ) )
2019ralbidva 2376 . . . . 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 ) ) ) )
21 ralcom 2530 . . . . 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 ) ) )
2220, 21syl6rbbr 197 . . . 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 452 . . 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 5024 . . 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 5024 . . 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 221 . 2  |-  ( H  Fn  A  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  `' R ,  `' S ( A ,  B ) ) )
273, 5, 26pm5.21nii 655 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  `' R ,  `' S ( A ,  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    e. wcel 1438   A.wral 2359   _Vcvv 2619   class class class wbr 3845   `'ccnv 4437    Fn wfn 5010   -1-1-onto->wf1o 5014   ` cfv 5015    Isom wiso 5016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-f1o 5022  df-fv 5023  df-isom 5024
This theorem is referenced by:  infisoti  6727
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