Theorem f1ocnvfvb 5848

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 5847	. . 3 |- ( ( F : A -1-1-onto-> B /\ C e. A ) -> ( ( F ` C ) = D -> ( `' F ` D ) = C ) )
2	1	3adant3 1019	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) -> ( ( F ` C ) = D -> ( `' F ` D ) = C ) )
3		fveq2 5575	. . . . 5 |- ( C = ( `' F ` D ) -> ( F ` C ) = ( F ` ( `' F ` D ) ) )
4	3	eqcoms 2207	. . . 4 |- ( ( `' F ` D ) = C -> ( F ` C ) = ( F ` ( `' F ` D ) ) )
5		f1ocnvfv2 5846	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ D e. B ) -> ( F ` ( `' F ` D ) ) = D )
6	5	eqeq2d 2216	. . . 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 1018	. 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 980   = wceq 1372   e. wcel 2175   `'ccnv 4673   -1-1-onto->wf1o 5269   ` cfv 5270

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278

This theorem is referenced by:  f1ofveu  5931  f1ocnvfv3  5932  seq3f1olemstep  10657  1arith2  12662  ennnfonelem1  12749  txhmeo  14762