Theorem f1ocnvfvb 5920

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 5919	. . 3 |- ( ( F : A -1-1-onto-> B /\ C e. A ) -> ( ( F ` C ) = D -> ( `' F ` D ) = C ) )
2	1	3adant3 1043	. 2 |- ( ( F : A -1-1-onto-> B /\ C e. A /\ D e. B ) -> ( ( F ` C ) = D -> ( `' F ` D ) = C ) )
3		fveq2 5639	. . . . 5 |- ( C = ( `' F ` D ) -> ( F ` C ) = ( F ` ( `' F ` D ) ) )
4	3	eqcoms 2234	. . . 4 |- ( ( `' F ` D ) = C -> ( F ` C ) = ( F ` ( `' F ` D ) ) )
5		f1ocnvfv2 5918	. . . . 5 |- ( ( F : A -1-1-onto-> B /\ D e. B ) -> ( F ` ( `' F ` D ) ) = D )
6	5	eqeq2d 2243	. . . 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 1042	. 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 1004   = wceq 1397   e. wcel 2202   `'ccnv 4724   -1-1-onto->wf1o 5325   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334

This theorem is referenced by:  f1ofveu  6005  f1ocnvfv3  6006  seq3f1olemstep  10775  1arith2  12940  ennnfonelem1  13027  txhmeo  15042