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Theorem f1ocnvfvrneq 6177
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 5827 . . 3  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
2 f1ocnv 5828 . . 3  |-  ( F : A -1-1-onto-> ran  F  ->  `' F : ran  F -1-1-onto-> A )
3 f1of1 5815 . . 3  |-  ( `' F : ran  F -1-1-onto-> A  ->  `' F : ran  F -1-1-> A )
4 f1veqaeq 6156 . . . 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 1379    e. wcel 1767   `'ccnv 4998   ran crn 5000   -1-1->wf1 5585   -1-1-onto->wf1o 5587   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596
This theorem is referenced by:  nbgraf1olem5  24149  usgra2adedgspthlem2  24316  constr3trllem2  24355
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