Theorem f1ocnvfvb 5953

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 5952	. . 3 |- ( ( F : A -1-1-onto-> B /\ C e. A ) -> ( ( F ` C ) = D -> ( `' F ` D ) = C ) )
2	1	3adant3 1044	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) -> ( ( F ` C ) = D -> ( `' F ` D ) = C ) )
3		fveq2 5670	. . . . 5 |- ( C = ( `' F ` D ) -> ( F ` C ) = ( F ` ( `' F ` D ) ) )
4	3	eqcoms 2235	. . . 4 |- ( ( `' F ` D ) = C -> ( F ` C ) = ( F ` ( `' F ` D ) ) )
5		f1ocnvfv2 5951	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ D e. B ) -> ( F ` ( `' F ` D ) ) = D )
6	5	eqeq2d 2244	. . . 4 |- ( ( F : A -1-1-onto-> B /\ D e. B ) -> ( ( F ` C ) = ( F ` ( `' F ` D ) ) <-> ( F ` C ) = D ) )
7	4, 6	imbitrid 154	. . 3 |- ( ( F : A -1-1-onto-> B /\ D e. B ) -> ( ( `' F ` D ) = C -> ( F ` C ) = D ) )
8	7	3adant2 1043	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) -> ( ( `' F ` D ) = C -> ( F ` C ) = D ) )
9	2, 8	impbid 129	1 |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) -> ( ( F ` C ) = D <-> ( `' F ` D ) = C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   `'ccnv 4748   -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-res 4761  df-ima 4762  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:  f1ofveu  6038  f1ocnvfv3  6039  seq3f1olemstep  10876  1arith2  13066  ennnfonelem1  13158  txhmeo  15184