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Theorem isocnv2 5681
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 5676 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
2 f1ofn 5336 . . 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 5676 . . 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 2663 . . . . . . . . . 10  |-  x  e. 
_V
7 vex 2663 . . . . . . . . . 10  |-  y  e. 
_V
86, 7brcnv 4692 . . . . . . . . 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 5406 . . . . . . . . . . 11  |-  ( ( Fun  H  /\  x  e.  dom  H )  -> 
( H `  x
)  e.  _V )
1110funfni 5193 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  x  e.  A )  ->  ( H `  x
)  e.  _V )
1211adantr 274 . . . . . . . . 9  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  ( H `  x )  e.  _V )
13 funfvex 5406 . . . . . . . . . . 11  |-  ( ( Fun  H  /\  y  e.  dom  H )  -> 
( H `  y
)  e.  _V )
1413funfni 5193 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  y  e.  A )  ->  ( H `  y
)  e.  _V )
1514adantlr 468 . . . . . . . . 9  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  ( H `  y )  e.  _V )
16 brcnvg 4690 . . . . . . . . 9  |-  ( ( ( H `  x
)  e.  _V  /\  ( H `  y )  e.  _V )  -> 
( ( H `  x ) `' S
( H `  y
)  <->  ( H `  y ) S ( H `  x ) ) )
1712, 15, 16syl2anc 408 . . . . . . . 8  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  (
( H `  x
) `' S ( H `  y )  <-> 
( H `  y
) S ( H `
 x ) ) )
189, 17bibi12d 234 . . . . . . 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 2410 . . . . . 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 2410 . . . . 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 2571 . . . . 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 198 . . . 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 459 . . 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 5102 . . 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 5102 . . 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 222 . 2  |-  ( H  Fn  A  ->  ( H  Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  `' R ,  `' S ( A ,  B ) ) )
273, 5, 26pm5.21nii 678 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  `' R ,  `' S ( A ,  B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 1465   A.wral 2393   _Vcvv 2660   class class class wbr 3899   `'ccnv 4508    Fn wfn 5088   -1-1-onto->wf1o 5092   ` cfv 5093    Isom wiso 5094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-f1o 5100  df-fv 5101  df-isom 5102
This theorem is referenced by:  infisoti  6887
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