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Theorem isocnv2 6240
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 ralcom 2937 . . . 4  |-  ( 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 ) ) )
2 vex 3034 . . . . . . 7  |-  x  e. 
_V
3 vex 3034 . . . . . . 7  |-  y  e. 
_V
42, 3brcnv 5022 . . . . . 6  |-  ( x `' R y  <->  y R x )
5 fvex 5889 . . . . . . 7  |-  ( H `
 x )  e. 
_V
6 fvex 5889 . . . . . . 7  |-  ( H `
 y )  e. 
_V
75, 6brcnv 5022 . . . . . 6  |-  ( ( H `  x ) `' S ( H `  y )  <->  ( H `  y ) S ( H `  x ) )
84, 7bibi12i 322 . . . . 5  |-  ( ( x `' R y  <-> 
( H `  x
) `' S ( H `  y ) )  <->  ( y R x  <->  ( H `  y ) S ( H `  x ) ) )
982ralbii 2824 . . . 4  |-  ( 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 ) ) )
101, 9bitr4i 260 . . 3  |-  ( 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
) ) )
1110anbi2i 708 . 2  |-  ( ( 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
) ) ) )
12 df-isom 5598 . 2  |-  ( 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 ) ) ) )
13 df-isom 5598 . 2  |-  ( 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
) ) ) )
1411, 12, 133bitr4i 285 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  `' R ,  `' S ( A ,  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376   A.wral 2756   class class class wbr 4395   `'ccnv 4838   -1-1-onto->wf1o 5588   ` cfv 5589    Isom wiso 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-cnv 4847  df-iota 5553  df-fv 5597  df-isom 5598
This theorem is referenced by:  infiso  8041  wofib  8078  leiso  12663  gtiso  28356
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