Theorem f1orescnv 5489

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 5486	. . 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 5301	. . . 4 |- ( Fun `' F -> `' ( F |` R ) = ( `' F |` ( F " R ) ) )
4		df-ima 4651	. . . . . 6 |- ( F " R ) = ran ( F |` R )
5		dff1o5 5482	. . . . . . 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 2232	. . . . 5 |- ( ( F |` R ) : R -1-1-onto-> P -> ( F " R ) = P )
8	7	reseq2d 4919	. . . 4 |- ( ( F |` R ) : R -1-1-onto-> P -> ( `' F |` ( F " R ) ) = ( `' F |` P ) )
9	3, 8	sylan9eq 2240	. . 3 |- ( ( Fun `' F /\ ( F |` R ) : R -1-1-onto-> P ) -> `' ( F |` R ) = ( `' F |` P ) )
10		f1oeq1 5461	. . 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 1363   `'ccnv 4637   ran crn 4639   |` cres 4640   "cima 4641   Fun wfun 5222   -1-1->wf1 5225   -1-1-onto->wf1o 5227

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235

This theorem is referenced by:  f1oresrab  5694