Theorem f1ocnvfvrneq 5773

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 5464	. . 3 |- ( F : A -1-1-> B -> F : A -1-1-onto-> ran F )
2		f1ocnv 5466	. . 3 |- ( F : A -1-1-onto-> ran F -> `' F : ran F -1-1-onto-> A )
3		f1of1 5452	. . 3 |- ( `' F : ran F -1-1-onto-> A -> `' F : ran F -1-1-> A )
4		f1veqaeq 5760	. . . 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 1353   e. wcel 2146   `'ccnv 4619   ran crn 4621   -1-1->wf1 5205   -1-1-onto->wf1o 5207   ` cfv 5208

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216

This theorem is referenced by: (None)