Theorem f1ocnvfvb 5499

Description: Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)

Assertion

Ref	Expression
f1ocnvfvb	|- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) -> ( ( F ` C ) = D <-> ( `' F ` D ) = C ) )

Proof of Theorem f1ocnvfvb

Step	Hyp	Ref	Expression
1		f1ocnvfv 5498	. . 3 |- ( ( F : A -1-1-onto-> B /\ C e. A ) -> ( ( F ` C ) = D -> ( `' F ` D ) = C ) )
2	1	3adant3 959	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) -> ( ( F ` C ) = D -> ( `' F ` D ) = C ) )
3		fveq2 5253	. . . . 5 |- ( C = ( `' F ` D ) -> ( F ` C ) = ( F ` ( `' F ` D ) ) )
4	3	eqcoms 2086	. . . 4 |- ( ( `' F ` D ) = C -> ( F ` C ) = ( F ` ( `' F ` D ) ) )
5		f1ocnvfv2 5497	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ D e. B ) -> ( F ` ( `' F ` D ) ) = D )
6	5	eqeq2d 2094	. . . 4 |- ( ( F : A -1-1-onto-> B /\ D e. B ) -> ( ( F ` C ) = ( F ` ( `' F ` D ) ) <-> ( F ` C ) = D ) )
7	4, 6	syl5ib 152	. . 3 |- ( ( F : A -1-1-onto-> B /\ D e. B ) -> ( ( `' F ` D ) = C -> ( F ` C ) = D ) )
8	7	3adant2 958	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) -> ( ( `' F ` D ) = C -> ( F ` C ) = D ) )
9	2, 8	impbid 127	1 |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) -> ( ( F ` C ) = D <-> ( `' F ` D ) = C ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 920   = wceq 1285   e. wcel 1434   `'ccnv 4400   -1-1-onto->wf1o 4968   ` cfv 4969

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977

This theorem is referenced by:  f1ofveu  5579  f1ocnvfv3  5580