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Theorem f1ocnvfvrneq 5473
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 5188 . . 3  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
2 f1ocnv 5190 . . 3  |-  ( F : A -1-1-onto-> ran  F  ->  `' F : ran  F -1-1-onto-> A )
3 f1of1 5176 . . 3  |-  ( `' F : ran  F -1-1-onto-> A  ->  `' F : ran  F -1-1-> A )
4 f1veqaeq 5460 . . . 4  |-  ( ( `' F : ran  F -1-1-> A  /\  ( C  e. 
ran  F  /\  D  e. 
ran  F ) )  ->  ( ( `' F `  C )  =  ( `' F `  D )  ->  C  =  D ) )
54ex 113 . . 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 122 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 102    = wceq 1285    e. wcel 1434   `'ccnv 4390   ran crn 4392   -1-1->wf1 4949   -1-1-onto->wf1o 4951   ` cfv 4952
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960
This theorem is referenced by: (None)
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