ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1ocnvfvrneq Unicode version

Theorem f1ocnvfvrneq 5651
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 5346 . . 3  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
2 f1ocnv 5348 . . 3  |-  ( F : A -1-1-onto-> ran  F  ->  `' F : ran  F -1-1-onto-> A )
3 f1of1 5334 . . 3  |-  ( `' F : ran  F -1-1-onto-> A  ->  `' F : ran  F -1-1-> A )
4 f1veqaeq 5638 . . . 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 1316    e. wcel 1465   `'ccnv 4508   ran crn 4510   -1-1->wf1 5090   -1-1-onto->wf1o 5092   ` cfv 5093
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator