Theorem f1ocnvfvrneq 5955

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 5625	. . 3 |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )
2		f1ocnv 5627	. . 3 |- ( F : A -1-1-onto-> ran F -> `' F : ran F -1-1-onto-> A )
3		f1of1 5613	. . 3 |- ( `' F : ran F -1-1-onto-> A -> `' F : ran F -1-1-> A )
4		f1veqaeq 5942	. . . 4 |- ( ( `' F : ran F -1-1-> A /\ ( C e. ran F /\ D e. ran F ) ) -> ( ( `' F ` C ) = ( `' F ` D ) -> C = D ) )
5	4	ex 115	. . 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 124	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 104   = wceq 1398   e. wcel 2203   `'ccnv 4748   ran crn 4750   -1-1->wf1 5349   -1-1-onto->wf1o 5351   ` cfv 5352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360

This theorem is referenced by: (None)