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Theorem f1opr 26391
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 5783 . 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 5525 . . . . . . . . 9  |-  ( v  =  <. r ,  s
>.  ->  ( F `  v )  =  ( F `  <. r ,  s >. )
)
3 df-ov 5861 . . . . . . . . 9  |-  ( r F s )  =  ( F `  <. r ,  s >. )
42, 3syl6eqr 2333 . . . . . . . 8  |-  ( v  =  <. r ,  s
>.  ->  ( F `  v )  =  ( r F s ) )
54eqeq1d 2291 . . . . . . 7  |-  ( v  =  <. r ,  s
>.  ->  ( ( F `
 v )  =  ( F `  w
)  <->  ( r F s )  =  ( F `  w ) ) )
6 eqeq1 2289 . . . . . . 7  |-  ( v  =  <. r ,  s
>.  ->  ( v  =  w  <->  <. r ,  s
>.  =  w )
)
75, 6imbi12d 311 . . . . . 6  |-  ( v  =  <. r ,  s
>.  ->  ( ( ( F `  v )  =  ( F `  w )  ->  v  =  w )  <->  ( (
r F s )  =  ( F `  w )  ->  <. r ,  s >.  =  w ) ) )
87ralbidv 2563 . . . . 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 4827 . . . 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 5525 . . . . . . . . 9  |-  ( w  =  <. t ,  u >.  ->  ( F `  w )  =  ( F `  <. t ,  u >. ) )
11 df-ov 5861 . . . . . . . . 9  |-  ( t F u )  =  ( F `  <. t ,  u >. )
1210, 11syl6eqr 2333 . . . . . . . 8  |-  ( w  =  <. t ,  u >.  ->  ( F `  w )  =  ( t F u ) )
1312eqeq2d 2294 . . . . . . 7  |-  ( w  =  <. t ,  u >.  ->  ( ( r F s )  =  ( F `  w
)  <->  ( r F s )  =  ( t F u ) ) )
14 eqeq2 2292 . . . . . . . 8  |-  ( w  =  <. t ,  u >.  ->  ( <. r ,  s >.  =  w  <->  <. r ,  s >.  =  <. t ,  u >. ) )
15 vex 2791 . . . . . . . . 9  |-  r  e. 
_V
16 vex 2791 . . . . . . . . 9  |-  s  e. 
_V
1715, 16opth 4245 . . . . . . . 8  |-  ( <.
r ,  s >.  =  <. t ,  u >.  <-> 
( r  =  t  /\  s  =  u ) )
1814, 17syl6bb 252 . . . . . . 7  |-  ( w  =  <. t ,  u >.  ->  ( <. r ,  s >.  =  w  <-> 
( r  =  t  /\  s  =  u ) ) )
1913, 18imbi12d 311 . . . . . 6  |-  ( w  =  <. t ,  u >.  ->  ( ( ( r F s )  =  ( F `  w )  ->  <. r ,  s >.  =  w )  <->  ( ( r F s )  =  ( t F u )  ->  ( r  =  t  /\  s  =  u ) ) ) )
2019ralxp 4827 . . . . 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 2569 . . . 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 240 . . 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 675 . 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 240 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623   A.wral 2543   <.cop 3643    X. cxp 4687   -->wf 5251   -1-1->wf1 5252   ` cfv 5255  (class class class)co 5858
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fv 5263  df-ov 5861
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