Theorem f1orescnv 5599

Description: The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)

Assertion

Ref	Expression
f1orescnv	|- ( ( Fun `' F /\ ( F |` R ) : R -1-1-onto-> P ) -> ( `' F |` P ) : P -1-1-onto-> R )

Proof of Theorem f1orescnv

Step	Hyp	Ref	Expression
1		f1ocnv 5596	. . 3 |- ( ( F |` R ) : R -1-1-onto-> P -> `' ( F |` R ) : P -1-1-onto-> R )
2	1	adantl 277	. 2 |- ( ( Fun `' F /\ ( F |` R ) : R -1-1-onto-> P ) -> `' ( F |` R ) : P -1-1-onto-> R )
3		funcnvres 5403	. . . 4 |- ( Fun `' F -> `' ( F |` R ) = ( `' F |` ( F " R ) ) )
4		df-ima 4738	. . . . . 6 |- ( F " R ) = ran ( F |` R )
5		dff1o5 5592	. . . . . . 7 |- ( ( F |` R ) : R -1-1-onto-> P <-> ( ( F |` R ) : R -1-1-> P /\ ran ( F |` R ) = P ) )
6	5	simprbi 275	. . . . . 6 |- ( ( F |` R ) : R -1-1-onto-> P -> ran ( F |` R ) = P )
7	4, 6	eqtrid 2276	. . . . 5 |- ( ( F |` R ) : R -1-1-onto-> P -> ( F " R ) = P )
8	7	reseq2d 5013	. . . 4 |- ( ( F |` R ) : R -1-1-onto-> P -> ( `' F |` ( F " R ) ) = ( `' F |` P ) )
9	3, 8	sylan9eq 2284	. . 3 |- ( ( Fun `' F /\ ( F |` R ) : R -1-1-onto-> P ) -> `' ( F |` R ) = ( `' F |` P ) )
10		f1oeq1 5571	. . 3 |- ( `' ( F |` R ) = ( `' F |` P ) -> ( `' ( F |` R ) : P -1-1-onto-> R <-> ( `' F |` P ) : P -1-1-onto-> R ) )
11	9, 10	syl 14	. 2 |- ( ( Fun `' F /\ ( F |` R ) : R -1-1-onto-> P ) -> ( `' ( F |` R ) : P -1-1-onto-> R <-> ( `' F |` P ) : P -1-1-onto-> R ) )
12	2, 11	mpbid 147	1 |- ( ( Fun `' F /\ ( F |` R ) : R -1-1-onto-> P ) -> ( `' F |` P ) : P -1-1-onto-> R )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   `'ccnv 4724   ran crn 4726   |` cres 4727   "cima 4728   Fun wfun 5320   -1-1->wf1 5323   -1-1-onto->wf1o 5325

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-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  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-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333

This theorem is referenced by:  f1oresrab  5812