Theorem f1ocnvfvrneq 5750

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 5443	. . 3 |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )
2		f1ocnv 5445	. . 3 |- ( F : A -1-1-onto-> ran F -> `' F : ran F -1-1-onto-> A )
3		f1of1 5431	. . 3 |- ( `' F : ran F -1-1-onto-> A -> `' F : ran F -1-1-> A )
4		f1veqaeq 5737	. . . 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 1343   e. wcel 2136   `'ccnv 4603   ran crn 4605   -1-1->wf1 5185   -1-1-onto->wf1o 5187   ` cfv 5188

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196

This theorem is referenced by: (None)