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Theorem f1opr 26110
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 5936 . 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 5661 . . . . . . . . 9  |-  ( v  =  <. r ,  s
>.  ->  ( F `  v )  =  ( F `  <. r ,  s >. )
)
3 df-ov 6016 . . . . . . . . 9  |-  ( r F s )  =  ( F `  <. r ,  s >. )
42, 3syl6eqr 2430 . . . . . . . 8  |-  ( v  =  <. r ,  s
>.  ->  ( F `  v )  =  ( r F s ) )
54eqeq1d 2388 . . . . . . 7  |-  ( v  =  <. r ,  s
>.  ->  ( ( F `
 v )  =  ( F `  w
)  <->  ( r F s )  =  ( F `  w ) ) )
6 eqeq1 2386 . . . . . . 7  |-  ( v  =  <. r ,  s
>.  ->  ( v  =  w  <->  <. r ,  s
>.  =  w )
)
75, 6imbi12d 312 . . . . . 6  |-  ( v  =  <. r ,  s
>.  ->  ( ( ( F `  v )  =  ( F `  w )  ->  v  =  w )  <->  ( (
r F s )  =  ( F `  w )  ->  <. r ,  s >.  =  w ) ) )
87ralbidv 2662 . . . . 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 4949 . . . 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 5661 . . . . . . . . 9  |-  ( w  =  <. t ,  u >.  ->  ( F `  w )  =  ( F `  <. t ,  u >. ) )
11 df-ov 6016 . . . . . . . . 9  |-  ( t F u )  =  ( F `  <. t ,  u >. )
1210, 11syl6eqr 2430 . . . . . . . 8  |-  ( w  =  <. t ,  u >.  ->  ( F `  w )  =  ( t F u ) )
1312eqeq2d 2391 . . . . . . 7  |-  ( w  =  <. t ,  u >.  ->  ( ( r F s )  =  ( F `  w
)  <->  ( r F s )  =  ( t F u ) ) )
14 eqeq2 2389 . . . . . . . 8  |-  ( w  =  <. t ,  u >.  ->  ( <. r ,  s >.  =  w  <->  <. r ,  s >.  =  <. t ,  u >. ) )
15 vex 2895 . . . . . . . . 9  |-  r  e. 
_V
16 vex 2895 . . . . . . . . 9  |-  s  e. 
_V
1715, 16opth 4369 . . . . . . . 8  |-  ( <.
r ,  s >.  =  <. t ,  u >.  <-> 
( r  =  t  /\  s  =  u ) )
1814, 17syl6bb 253 . . . . . . 7  |-  ( w  =  <. t ,  u >.  ->  ( <. r ,  s >.  =  w  <-> 
( r  =  t  /\  s  =  u ) ) )
1913, 18imbi12d 312 . . . . . 6  |-  ( w  =  <. t ,  u >.  ->  ( ( ( r F s )  =  ( F `  w )  ->  <. r ,  s >.  =  w )  <->  ( ( r F s )  =  ( t F u )  ->  ( r  =  t  /\  s  =  u ) ) ) )
2019ralxp 4949 . . . . 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 2668 . . . 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 241 . . 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 676 . 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 241 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 177    /\ wa 359    = wceq 1649   A.wral 2642   <.cop 3753    X. cxp 4809   -->wf 5383   -1-1->wf1 5384   ` cfv 5387  (class class class)co 6013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fv 5395  df-ov 6016
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