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Theorem isocnv2 5791
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 5786 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
2 f1ofn 5443 . . 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 5786 . . 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 2633 . . . . 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 2733 . . . . . . . . . 10  |-  x  e. 
_V
8 vex 2733 . . . . . . . . . 10  |-  y  e. 
_V
97, 8brcnv 4794 . . . . . . . . 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 5513 . . . . . . . . . . 11  |-  ( ( Fun  H  /\  x  e.  dom  H )  -> 
( H `  x
)  e.  _V )
1211funfni 5298 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  x  e.  A )  ->  ( H `  x
)  e.  _V )
1312adantr 274 . . . . . . . . 9  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  ( H `  x )  e.  _V )
14 funfvex 5513 . . . . . . . . . . 11  |-  ( ( Fun  H  /\  y  e.  dom  H )  -> 
( H `  y
)  e.  _V )
1514funfni 5298 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  y  e.  A )  ->  ( H `  y
)  e.  _V )
1615adantlr 474 . . . . . . . . 9  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  ( H `  y )  e.  _V )
17 brcnvg 4792 . . . . . . . . 9  |-  ( ( ( H `  x
)  e.  _V  /\  ( H `  y )  e.  _V )  -> 
( ( H `  x ) `' S
( H `  y
)  <->  ( H `  y ) S ( H `  x ) ) )
1813, 16, 17syl2anc 409 . . . . . . . 8  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  (
( H `  x
) `' S ( H `  y )  <-> 
( H `  y
) S ( H `
 x ) ) )
1910, 18bibi12d 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 ) ) ) )
2019ralbidva 2466 . . . . . 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 2466 . . . . 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 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 461 . . 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 5207 . . 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 5207 . . 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 699 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 2141   A.wral 2448   _Vcvv 2730   class class class wbr 3989   `'ccnv 4610    Fn wfn 5193   -1-1-onto->wf1o 5197   ` cfv 5198    Isom wiso 5199
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-f1o 5205  df-fv 5206  df-isom 5207
This theorem is referenced by:  infisoti  7009
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