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Theorem f1veqaeq 5791
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 5790 . . 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 5534 . . . . . . . 8  |-  ( c  =  C  ->  ( F `  c )  =  ( F `  C ) )
32eqeq1d 2198 . . . . . . 7  |-  ( c  =  C  ->  (
( F `  c
)  =  ( F `
 d )  <->  ( F `  C )  =  ( F `  d ) ) )
4 eqeq1 2196 . . . . . . 7  |-  ( c  =  C  ->  (
c  =  d  <->  C  =  d ) )
53, 4imbi12d 234 . . . . . 6  |-  ( c  =  C  ->  (
( ( F `  c )  =  ( F `  d )  ->  c  =  d )  <->  ( ( F `
 C )  =  ( F `  d
)  ->  C  =  d ) ) )
6 fveq2 5534 . . . . . . . 8  |-  ( d  =  D  ->  ( F `  d )  =  ( F `  D ) )
76eqeq2d 2201 . . . . . . 7  |-  ( d  =  D  ->  (
( F `  C
)  =  ( F `
 d )  <->  ( F `  C )  =  ( F `  D ) ) )
8 eqeq2 2199 . . . . . . 7  |-  ( d  =  D  ->  ( C  =  d  <->  C  =  D ) )
97, 8imbi12d 234 . . . . . 6  |-  ( d  =  D  ->  (
( ( F `  C )  =  ( F `  d )  ->  C  =  d )  <->  ( ( F `
 C )  =  ( F `  D
)  ->  C  =  D ) ) )
105, 9rspc2v 2869 . . . . 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 277 . . 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 121 . 2  |-  ( F : A -1-1-> B  -> 
( ( C  e.  A  /\  D  e.  A )  ->  (
( F `  C
)  =  ( F `
 D )  ->  C  =  D )
) )
1413imp 124 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 104    = wceq 1364    e. wcel 2160   A.wral 2468   -->wf 5231   -1-1->wf1 5232   ` cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fv 5243
This theorem is referenced by:  f1fveq  5794  f1ocnvfvrneq  5804  f1o2ndf1  6254  fidceq  6898  difinfsnlem  7129  difinfsn  7130  iseqf1olemab  10522  iseqf1olemnanb  10523  f1ghm0to0  13228  1dom1el  15221  pwle2  15227
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