Theorem f1ocnvfvb 5681

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 5680	. . 3 |- ( ( F : A -1-1-onto-> B /\ C e. A ) -> ( ( F ` C ) = D -> ( `' F ` D ) = C ) )
2	1	3adant3 1001	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) -> ( ( F ` C ) = D -> ( `' F ` D ) = C ) )
3		fveq2 5421	. . . . 5 |- ( C = ( `' F ` D ) -> ( F ` C ) = ( F ` ( `' F ` D ) ) )
4	3	eqcoms 2142	. . . 4 |- ( ( `' F ` D ) = C -> ( F ` C ) = ( F ` ( `' F ` D ) ) )
5		f1ocnvfv2 5679	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ D e. B ) -> ( F ` ( `' F ` D ) ) = D )
6	5	eqeq2d 2151	. . . 4 |- ( ( F : A -1-1-onto-> B /\ D e. B ) -> ( ( F ` C ) = ( F ` ( `' F ` D ) ) <-> ( F ` C ) = D ) )
7	4, 6	syl5ib 153	. . 3 |- ( ( F : A -1-1-onto-> B /\ D e. B ) -> ( ( `' F ` D ) = C -> ( F ` C ) = D ) )
8	7	3adant2 1000	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) -> ( ( `' F ` D ) = C -> ( F ` C ) = D ) )
9	2, 8	impbid 128	1 |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) -> ( ( F ` C ) = D <-> ( `' F ` D ) = C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   `'ccnv 4538   -1-1-onto->wf1o 5122   ` cfv 5123

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131

This theorem is referenced by:  f1ofveu  5762  f1ocnvfv3  5763  seq3f1olemstep  10274  ennnfonelem1  11920  txhmeo  12488