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Theorem isocnv2 6202
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 3015 . . . 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 3109 . . . . . . 7  |-  x  e. 
_V
3 vex 3109 . . . . . . 7  |-  y  e. 
_V
42, 3brcnv 5174 . . . . . 6  |-  ( x `' R y  <->  y R x )
5 fvex 5858 . . . . . . 7  |-  ( H `
 x )  e. 
_V
6 fvex 5858 . . . . . . 7  |-  ( H `
 y )  e. 
_V
75, 6brcnv 5174 . . . . . 6  |-  ( ( H `  x ) `' S ( H `  y )  <->  ( H `  y ) S ( H `  x ) )
84, 7bibi12i 313 . . . . 5  |-  ( ( x `' R y  <-> 
( H `  x
) `' S ( H `  y ) )  <->  ( y R x  <->  ( H `  y ) S ( H `  x ) ) )
982ralbii 2886 . . . 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 252 . . 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 692 . 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 5579 . 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 5579 . 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 277 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  <-> 
H  Isom  `' R ,  `' S ( A ,  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367   A.wral 2804   class class class wbr 4439   `'ccnv 4987   -1-1-onto->wf1o 5569   ` cfv 5570    Isom wiso 5571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-cnv 4996  df-iota 5534  df-fv 5578  df-isom 5579
This theorem is referenced by:  wofib  7962  leiso  12492  gtiso  27747
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