Theorem f1orescnv 5655

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 5652	. . 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 5454	. . . 4 |- ( Fun `' F -> `' ( F |` R ) = ( `' F |` ( F " R ) ) )
4		df-ima 4787	. . . . . 6 |- ( F " R ) = ran ( F |` R )
5		dff1o5 5648	. . . . . . 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 2283	. . . . 5 |- ( ( F |` R ) : R -1-1-onto-> P -> ( F " R ) = P )
8	7	reseq2d 5063	. . . 4 |- ( ( F |` R ) : R -1-1-onto-> P -> ( `' F |` ( F " R ) ) = ( `' F |` P ) )
9	3, 8	sylan9eq 2291	. . 3 |- ( ( Fun `' F /\ ( F |` R ) : R -1-1-onto-> P ) -> `' ( F |` R ) = ( `' F |` P ) )
10		f1oeq1 5627	. . 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   `'ccnv 4773   ran crn 4775   |` cres 4776   "cima 4777   Fun wfun 5371   -1-1->wf1 5374   -1-1-onto->wf1o 5376

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384

This theorem is used by:  f1oresrab  5873