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Theorem isocnv2 6222
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 2951 . . . 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 3048 . . . . . . 7  |-  x  e. 
_V
3 vex 3048 . . . . . . 7  |-  y  e. 
_V
42, 3brcnv 5017 . . . . . 6  |-  ( x `' R y  <->  y R x )
5 fvex 5875 . . . . . . 7  |-  ( H `
 x )  e. 
_V
6 fvex 5875 . . . . . . 7  |-  ( H `
 y )  e. 
_V
75, 6brcnv 5017 . . . . . 6  |-  ( ( H `  x ) `' S ( H `  y )  <->  ( H `  y ) S ( H `  x ) )
84, 7bibi12i 317 . . . . 5  |-  ( ( x `' R y  <-> 
( H `  x
) `' S ( H `  y ) )  <->  ( y R x  <->  ( H `  y ) S ( H `  x ) ) )
982ralbii 2820 . . . 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 256 . . 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 700 . 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 5591 . 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 5591 . 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 281 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  `' R ,  `' S ( A ,  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371   A.wral 2737   class class class wbr 4402   `'ccnv 4833   -1-1-onto->wf1o 5581   ` cfv 5582    Isom wiso 5583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-cnv 4842  df-iota 5546  df-fv 5590  df-isom 5591
This theorem is referenced by:  infiso  8023  wofib  8060  leiso  12622  gtiso  28281
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