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Theorem f1ocnvfvrneq 5676
Description: If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
Assertion
Ref Expression
f1ocnvfvrneq  |-  ( ( F : A -1-1-> B  /\  ( C  e.  ran  F  /\  D  e.  ran  F ) )  ->  (
( `' F `  C )  =  ( `' F `  D )  ->  C  =  D ) )

Proof of Theorem f1ocnvfvrneq
StepHypRef Expression
1 f1f1orn 5371 . . 3  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
2 f1ocnv 5373 . . 3  |-  ( F : A -1-1-onto-> ran  F  ->  `' F : ran  F -1-1-onto-> A )
3 f1of1 5359 . . 3  |-  ( `' F : ran  F -1-1-onto-> A  ->  `' F : ran  F -1-1-> A )
4 f1veqaeq 5663 . . . 4  |-  ( ( `' F : ran  F -1-1-> A  /\  ( C  e. 
ran  F  /\  D  e. 
ran  F ) )  ->  ( ( `' F `  C )  =  ( `' F `  D )  ->  C  =  D ) )
54ex 114 . . 3  |-  ( `' F : ran  F -1-1-> A  ->  ( ( C  e.  ran  F  /\  D  e.  ran  F )  ->  ( ( `' F `  C )  =  ( `' F `  D )  ->  C  =  D ) ) )
61, 2, 3, 54syl 18 . 2  |-  ( F : A -1-1-> B  -> 
( ( C  e. 
ran  F  /\  D  e. 
ran  F )  -> 
( ( `' F `  C )  =  ( `' F `  D )  ->  C  =  D ) ) )
76imp 123 1  |-  ( ( F : A -1-1-> B  /\  ( C  e.  ran  F  /\  D  e.  ran  F ) )  ->  (
( `' F `  C )  =  ( `' F `  D )  ->  C  =  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   `'ccnv 4533   ran crn 4535   -1-1->wf1 5115   -1-1-onto->wf1o 5117   ` 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-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126
This theorem is referenced by: (None)
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