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Theorem f1ocnvfvrneq 5995
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 5657 . . 3  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
2 f1ocnv 5658 . . 3  |-  ( F : A -1-1-onto-> ran  F  ->  `' F : ran  F -1-1-onto-> A )
3 f1of1 5645 . . 3  |-  ( `' F : ran  F -1-1-onto-> A  ->  `' F : ran  F -1-1-> A )
4 f1veqaeq 5977 . . . 4  |-  ( ( `' F : ran  F -1-1-> A  /\  ( C  e. 
ran  F  /\  D  e. 
ran  F ) )  ->  ( ( `' F `  C )  =  ( `' F `  D )  ->  C  =  D ) )
54ex 434 . . 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 21 . 2  |-  ( F : A -1-1-> B  -> 
( ( C  e. 
ran  F  /\  D  e. 
ran  F )  -> 
( ( `' F `  C )  =  ( `' F `  D )  ->  C  =  D ) ) )
76imp 429 1  |-  ( ( F : A -1-1-> B  /\  ( C  e.  ran  F  /\  D  e.  ran  F ) )  ->  (
( `' F `  C )  =  ( `' F `  D )  ->  C  =  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   `'ccnv 4844   ran crn 4846   -1-1->wf1 5420   -1-1-onto->wf1o 5422   ` cfv 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431
This theorem is referenced by:  nbgraf1olem5  23359  constr3trllem2  23542  usgra2adedgspthlem2  30309
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