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Theorem f1veqaeq 5548
Description: If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
Assertion
Ref Expression
f1veqaeq  |-  ( ( F : A -1-1-> B  /\  ( C  e.  A  /\  D  e.  A
) )  ->  (
( F `  C
)  =  ( F `
 D )  ->  C  =  D )
)

Proof of Theorem f1veqaeq
Dummy variables  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dff13 5547 . . 3  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  A. c  e.  A  A. d  e.  A  (
( F `  c
)  =  ( F `
 d )  -> 
c  =  d ) ) )
2 fveq2 5305 . . . . . . . 8  |-  ( c  =  C  ->  ( F `  c )  =  ( F `  C ) )
32eqeq1d 2096 . . . . . . 7  |-  ( c  =  C  ->  (
( F `  c
)  =  ( F `
 d )  <->  ( F `  C )  =  ( F `  d ) ) )
4 eqeq1 2094 . . . . . . 7  |-  ( c  =  C  ->  (
c  =  d  <->  C  =  d ) )
53, 4imbi12d 232 . . . . . 6  |-  ( c  =  C  ->  (
( ( F `  c )  =  ( F `  d )  ->  c  =  d )  <->  ( ( F `
 C )  =  ( F `  d
)  ->  C  =  d ) ) )
6 fveq2 5305 . . . . . . . 8  |-  ( d  =  D  ->  ( F `  d )  =  ( F `  D ) )
76eqeq2d 2099 . . . . . . 7  |-  ( d  =  D  ->  (
( F `  C
)  =  ( F `
 d )  <->  ( F `  C )  =  ( F `  D ) ) )
8 eqeq2 2097 . . . . . . 7  |-  ( d  =  D  ->  ( C  =  d  <->  C  =  D ) )
97, 8imbi12d 232 . . . . . 6  |-  ( d  =  D  ->  (
( ( F `  C )  =  ( F `  d )  ->  C  =  d )  <->  ( ( F `
 C )  =  ( F `  D
)  ->  C  =  D ) ) )
105, 9rspc2v 2734 . . . . 5  |-  ( ( C  e.  A  /\  D  e.  A )  ->  ( A. c  e.  A  A. d  e.  A  ( ( F `
 c )  =  ( F `  d
)  ->  c  =  d )  ->  (
( F `  C
)  =  ( F `
 D )  ->  C  =  D )
) )
1110com12 30 . . . 4  |-  ( A. c  e.  A  A. d  e.  A  (
( F `  c
)  =  ( F `
 d )  -> 
c  =  d )  ->  ( ( C  e.  A  /\  D  e.  A )  ->  (
( F `  C
)  =  ( F `
 D )  ->  C  =  D )
) )
1211adantl 271 . . 3  |-  ( ( F : A --> B  /\  A. c  e.  A  A. d  e.  A  (
( F `  c
)  =  ( F `
 d )  -> 
c  =  d ) )  ->  ( ( C  e.  A  /\  D  e.  A )  ->  ( ( F `  C )  =  ( F `  D )  ->  C  =  D ) ) )
131, 12sylbi 119 . 2  |-  ( F : A -1-1-> B  -> 
( ( C  e.  A  /\  D  e.  A )  ->  (
( F `  C
)  =  ( F `
 D )  ->  C  =  D )
) )
1413imp 122 1  |-  ( ( F : A -1-1-> B  /\  ( C  e.  A  /\  D  e.  A
) )  ->  (
( F `  C
)  =  ( F `
 D )  ->  C  =  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   A.wral 2359   -->wf 5011   -1-1->wf1 5012   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fv 5023
This theorem is referenced by:  f1fveq  5551  f1ocnvfvrneq  5561  f1o2ndf1  5993  fidceq  6585  iseqf1olemab  9918  iseqf1olemnanb  9919
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