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Theorem f1opr 29816
Description: Condition for an operation to be one-to-one. (Contributed by Jeff Madsen, 17-Jun-2010.)
Assertion
Ref Expression
f1opr  |-  ( F : ( A  X.  B ) -1-1-> C  <->  ( F : ( A  X.  B ) --> C  /\  A. r  e.  A  A. s  e.  B  A. t  e.  A  A. u  e.  B  (
( r F s )  =  ( t F u )  -> 
( r  =  t  /\  s  =  u ) ) ) )
Distinct variable groups:    A, r,
s, t, u    B, r, s, t, u    F, r, s, t, u
Allowed substitution hints:    C( u, t, s, r)

Proof of Theorem f1opr
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dff13 6152 . 2  |-  ( F : ( A  X.  B ) -1-1-> C  <->  ( F : ( A  X.  B ) --> C  /\  A. v  e.  ( A  X.  B ) A. w  e.  ( A  X.  B ) ( ( F `  v )  =  ( F `  w )  ->  v  =  w ) ) )
2 fveq2 5864 . . . . . . . . 9  |-  ( v  =  <. r ,  s
>.  ->  ( F `  v )  =  ( F `  <. r ,  s >. )
)
3 df-ov 6285 . . . . . . . . 9  |-  ( r F s )  =  ( F `  <. r ,  s >. )
42, 3syl6eqr 2526 . . . . . . . 8  |-  ( v  =  <. r ,  s
>.  ->  ( F `  v )  =  ( r F s ) )
54eqeq1d 2469 . . . . . . 7  |-  ( v  =  <. r ,  s
>.  ->  ( ( F `
 v )  =  ( F `  w
)  <->  ( r F s )  =  ( F `  w ) ) )
6 eqeq1 2471 . . . . . . 7  |-  ( v  =  <. r ,  s
>.  ->  ( v  =  w  <->  <. r ,  s
>.  =  w )
)
75, 6imbi12d 320 . . . . . 6  |-  ( v  =  <. r ,  s
>.  ->  ( ( ( F `  v )  =  ( F `  w )  ->  v  =  w )  <->  ( (
r F s )  =  ( F `  w )  ->  <. r ,  s >.  =  w ) ) )
87ralbidv 2903 . . . . 5  |-  ( v  =  <. r ,  s
>.  ->  ( A. w  e.  ( A  X.  B
) ( ( F `
 v )  =  ( F `  w
)  ->  v  =  w )  <->  A. w  e.  ( A  X.  B
) ( ( r F s )  =  ( F `  w
)  ->  <. r ,  s >.  =  w
) ) )
98ralxp 5142 . . . 4  |-  ( A. v  e.  ( A  X.  B ) A. w  e.  ( A  X.  B
) ( ( F `
 v )  =  ( F `  w
)  ->  v  =  w )  <->  A. r  e.  A  A. s  e.  B  A. w  e.  ( A  X.  B
) ( ( r F s )  =  ( F `  w
)  ->  <. r ,  s >.  =  w
) )
10 fveq2 5864 . . . . . . . . 9  |-  ( w  =  <. t ,  u >.  ->  ( F `  w )  =  ( F `  <. t ,  u >. ) )
11 df-ov 6285 . . . . . . . . 9  |-  ( t F u )  =  ( F `  <. t ,  u >. )
1210, 11syl6eqr 2526 . . . . . . . 8  |-  ( w  =  <. t ,  u >.  ->  ( F `  w )  =  ( t F u ) )
1312eqeq2d 2481 . . . . . . 7  |-  ( w  =  <. t ,  u >.  ->  ( ( r F s )  =  ( F `  w
)  <->  ( r F s )  =  ( t F u ) ) )
14 eqeq2 2482 . . . . . . . 8  |-  ( w  =  <. t ,  u >.  ->  ( <. r ,  s >.  =  w  <->  <. r ,  s >.  =  <. t ,  u >. ) )
15 vex 3116 . . . . . . . . 9  |-  r  e. 
_V
16 vex 3116 . . . . . . . . 9  |-  s  e. 
_V
1715, 16opth 4721 . . . . . . . 8  |-  ( <.
r ,  s >.  =  <. t ,  u >.  <-> 
( r  =  t  /\  s  =  u ) )
1814, 17syl6bb 261 . . . . . . 7  |-  ( w  =  <. t ,  u >.  ->  ( <. r ,  s >.  =  w  <-> 
( r  =  t  /\  s  =  u ) ) )
1913, 18imbi12d 320 . . . . . 6  |-  ( w  =  <. t ,  u >.  ->  ( ( ( r F s )  =  ( F `  w )  ->  <. r ,  s >.  =  w )  <->  ( ( r F s )  =  ( t F u )  ->  ( r  =  t  /\  s  =  u ) ) ) )
2019ralxp 5142 . . . . 5  |-  ( A. w  e.  ( A  X.  B ) ( ( r F s )  =  ( F `  w )  ->  <. r ,  s >.  =  w )  <->  A. t  e.  A  A. u  e.  B  ( ( r F s )  =  ( t F u )  ->  ( r  =  t  /\  s  =  u ) ) )
21202ralbii 2896 . . . 4  |-  ( A. r  e.  A  A. s  e.  B  A. w  e.  ( A  X.  B ) ( ( r F s )  =  ( F `  w )  ->  <. r ,  s >.  =  w )  <->  A. r  e.  A  A. s  e.  B  A. t  e.  A  A. u  e.  B  ( ( r F s )  =  ( t F u )  ->  ( r  =  t  /\  s  =  u ) ) )
229, 21bitri 249 . . 3  |-  ( A. v  e.  ( A  X.  B ) A. w  e.  ( A  X.  B
) ( ( F `
 v )  =  ( F `  w
)  ->  v  =  w )  <->  A. r  e.  A  A. s  e.  B  A. t  e.  A  A. u  e.  B  ( (
r F s )  =  ( t F u )  ->  (
r  =  t  /\  s  =  u )
) )
2322anbi2i 694 . 2  |-  ( ( F : ( A  X.  B ) --> C  /\  A. v  e.  ( A  X.  B
) A. w  e.  ( A  X.  B
) ( ( F `
 v )  =  ( F `  w
)  ->  v  =  w ) )  <->  ( F : ( A  X.  B ) --> C  /\  A. r  e.  A  A. s  e.  B  A. t  e.  A  A. u  e.  B  (
( r F s )  =  ( t F u )  -> 
( r  =  t  /\  s  =  u ) ) ) )
241, 23bitri 249 1  |-  ( F : ( A  X.  B ) -1-1-> C  <->  ( F : ( A  X.  B ) --> C  /\  A. r  e.  A  A. s  e.  B  A. t  e.  A  A. u  e.  B  (
( r F s )  =  ( t F u )  -> 
( r  =  t  /\  s  =  u ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   A.wral 2814   <.cop 4033    X. cxp 4997   -->wf 5582   -1-1->wf1 5583   ` cfv 5586  (class class class)co 6282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fv 5594  df-ov 6285
This theorem is referenced by: (None)
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