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Theorem f1opr 28615
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 5969 . 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 5689 . . . . . . . . 9  |-  ( v  =  <. r ,  s
>.  ->  ( F `  v )  =  ( F `  <. r ,  s >. )
)
3 df-ov 6092 . . . . . . . . 9  |-  ( r F s )  =  ( F `  <. r ,  s >. )
42, 3syl6eqr 2491 . . . . . . . 8  |-  ( v  =  <. r ,  s
>.  ->  ( F `  v )  =  ( r F s ) )
54eqeq1d 2449 . . . . . . 7  |-  ( v  =  <. r ,  s
>.  ->  ( ( F `
 v )  =  ( F `  w
)  <->  ( r F s )  =  ( F `  w ) ) )
6 eqeq1 2447 . . . . . . 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 2733 . . . . 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 4979 . . . 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 5689 . . . . . . . . 9  |-  ( w  =  <. t ,  u >.  ->  ( F `  w )  =  ( F `  <. t ,  u >. ) )
11 df-ov 6092 . . . . . . . . 9  |-  ( t F u )  =  ( F `  <. t ,  u >. )
1210, 11syl6eqr 2491 . . . . . . . 8  |-  ( w  =  <. t ,  u >.  ->  ( F `  w )  =  ( t F u ) )
1312eqeq2d 2452 . . . . . . 7  |-  ( w  =  <. t ,  u >.  ->  ( ( r F s )  =  ( F `  w
)  <->  ( r F s )  =  ( t F u ) ) )
14 eqeq2 2450 . . . . . . . 8  |-  ( w  =  <. t ,  u >.  ->  ( <. r ,  s >.  =  w  <->  <. r ,  s >.  =  <. t ,  u >. ) )
15 vex 2973 . . . . . . . . 9  |-  r  e. 
_V
16 vex 2973 . . . . . . . . 9  |-  s  e. 
_V
1715, 16opth 4564 . . . . . . . 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 4979 . . . . 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 2739 . . . 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 1369   A.wral 2713   <.cop 3881    X. cxp 4836   -->wf 5412   -1-1->wf1 5413   ` cfv 5416  (class class class)co 6089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fv 5424  df-ov 6092
This theorem is referenced by: (None)
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