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Theorem isocnv2 5870
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 2734 . . . 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 2825 . . . . . . 7  |-  x  e. 
_V
3 vex 2825 . . . . . . 7  |-  y  e. 
_V
42, 3brcnv 4901 . . . . . 6  |-  ( x `' R y  <->  y R x )
5 fvex 5577 . . . . . . 7  |-  ( H `
 x )  e. 
_V
6 fvex 5577 . . . . . . 7  |-  ( H `
 y )  e. 
_V
75, 6brcnv 4901 . . . . . 6  |-  ( ( H `  x ) `' S ( H `  y )  <->  ( H `  y ) S ( H `  x ) )
84, 7bibi12i 306 . . . . 5  |-  ( ( x `' R y  <-> 
( H `  x
) `' S ( H `  y ) )  <->  ( y R x  <->  ( H `  y ) S ( H `  x ) ) )
982ralbii 2603 . . . 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 243 . . 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 675 . 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 5301 . 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 5301 . 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 268 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  `' R ,  `' S ( A ,  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   A.wral 2577   class class class wbr 4060   `'ccnv 4725   -1-1-onto->wf1o 5291   ` cfv 5292    Isom wiso 5293
This theorem is referenced by:  wofib  7305  leiso  11444  gtiso  23238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-cnv 4734  df-iota 5256  df-fv 5300  df-isom 5301
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