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Theorem f1veqaeq 5663
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 5662 . . 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 5414 . . . . . . . 8  |-  ( c  =  C  ->  ( F `  c )  =  ( F `  C ) )
32eqeq1d 2146 . . . . . . 7  |-  ( c  =  C  ->  (
( F `  c
)  =  ( F `
 d )  <->  ( F `  C )  =  ( F `  d ) ) )
4 eqeq1 2144 . . . . . . 7  |-  ( c  =  C  ->  (
c  =  d  <->  C  =  d ) )
53, 4imbi12d 233 . . . . . 6  |-  ( c  =  C  ->  (
( ( F `  c )  =  ( F `  d )  ->  c  =  d )  <->  ( ( F `
 C )  =  ( F `  d
)  ->  C  =  d ) ) )
6 fveq2 5414 . . . . . . . 8  |-  ( d  =  D  ->  ( F `  d )  =  ( F `  D ) )
76eqeq2d 2149 . . . . . . 7  |-  ( d  =  D  ->  (
( F `  C
)  =  ( F `
 d )  <->  ( F `  C )  =  ( F `  D ) ) )
8 eqeq2 2147 . . . . . . 7  |-  ( d  =  D  ->  ( C  =  d  <->  C  =  D ) )
97, 8imbi12d 233 . . . . . 6  |-  ( d  =  D  ->  (
( ( F `  C )  =  ( F `  d )  ->  C  =  d )  <->  ( ( F `
 C )  =  ( F `  D
)  ->  C  =  D ) ) )
105, 9rspc2v 2797 . . . . 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 275 . . 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 120 . 2  |-  ( F : A -1-1-> B  -> 
( ( C  e.  A  /\  D  e.  A )  ->  (
( F `  C
)  =  ( F `
 D )  ->  C  =  D )
) )
1413imp 123 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 103    = wceq 1331    e. wcel 1480   A.wral 2414   -->wf 5114   -1-1->wf1 5115   ` cfv 5118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fv 5126
This theorem is referenced by:  f1fveq  5666  f1ocnvfvrneq  5676  f1o2ndf1  6118  fidceq  6756  difinfsnlem  6977  difinfsn  6978  iseqf1olemab  10255  iseqf1olemnanb  10256  pwle2  13182
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