Theorem f1ocnvfvrneq 5575

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

Step	Hyp	Ref	Expression
1		f1f1orn 5277	. . 3 |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )
2		f1ocnv 5279	. . 3 |- ( F : A -1-1-onto-> ran F -> `' F : ran F -1-1-onto-> A )
3		f1of1 5265	. . 3 |- ( `' F : ran F -1-1-onto-> A -> `' F : ran F -1-1-> A )
4		f1veqaeq 5562	. . . 4 |- ( ( `' F : ran F -1-1-> A /\ ( C e. ran F /\ D e. ran F ) ) -> ( ( `' F ` C ) = ( `' F ` D ) -> C = D ) )
5	4	ex 114	. . 3 |- ( `' F : ran F -1-1-> A -> ( ( C e. ran F /\ D e. ran F ) -> ( ( `' F ` C ) = ( `' F ` D ) -> C = D ) ) )
6	1, 2, 3, 5	4syl 18	. 2 |- ( F : A -1-1-> B -> ( ( C e. ran F /\ D e. ran F ) -> ( ( `' F ` C ) = ( `' F ` D ) -> C = D ) ) )
7	6	imp 123	1 |- ( ( F : A -1-1-> B /\ ( C e. ran F /\ D e. ran F ) ) -> ( ( `' F ` C ) = ( `' F ` D ) -> C = D ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   `'ccnv 4450   ran crn 4452   -1-1->wf1 5025   -1-1-onto->wf1o 5027   ` cfv 5028

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036

This theorem is referenced by: (None)