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Theorem isocnv2 5762
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 5757 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
2 f1ofn 5415 . . 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 5757 . . 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 2620 . . . . 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 2715 . . . . . . . . . 10  |-  x  e. 
_V
8 vex 2715 . . . . . . . . . 10  |-  y  e. 
_V
97, 8brcnv 4769 . . . . . . . . 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 5485 . . . . . . . . . . 11  |-  ( ( Fun  H  /\  x  e.  dom  H )  -> 
( H `  x
)  e.  _V )
1211funfni 5270 . . . . . . . . . 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 5485 . . . . . . . . . . 11  |-  ( ( Fun  H  /\  y  e.  dom  H )  -> 
( H `  y
)  e.  _V )
1514funfni 5270 . . . . . . . . . 10  |-  ( ( H  Fn  A  /\  y  e.  A )  ->  ( H `  y
)  e.  _V )
1615adantlr 469 . . . . . . . . 9  |-  ( ( ( H  Fn  A  /\  x  e.  A
)  /\  y  e.  A )  ->  ( H `  y )  e.  _V )
17 brcnvg 4767 . . . . . . . . 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 2453 . . . . . 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 2453 . . . . 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 460 . . 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 5179 . . 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 5179 . . 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 694 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 2128   A.wral 2435   _Vcvv 2712   class class class wbr 3965   `'ccnv 4585    Fn wfn 5165   -1-1-onto->wf1o 5169   ` cfv 5170    Isom wiso 5171
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-f1o 5177  df-fv 5178  df-isom 5179
This theorem is referenced by:  infisoti  6976
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