Theorem f1orescnv 5635

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 5632	. . 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 5434	. . . 4 |- ( Fun `' F -> `' ( F |` R ) = ( `' F |` ( F " R ) ) )
4		df-ima 4767	. . . . . 6 |- ( F " R ) = ran ( F |` R )
5		dff1o5 5628	. . . . . . 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 2279	. . . . 5 |- ( ( F |` R ) : R -1-1-onto-> P -> ( F " R ) = P )
8	7	reseq2d 5043	. . . 4 |- ( ( F |` R ) : R -1-1-onto-> P -> ( `' F |` ( F " R ) ) = ( `' F |` P ) )
9	3, 8	sylan9eq 2287	. . 3 |- ( ( Fun `' F /\ ( F |` R ) : R -1-1-onto-> P ) -> `' ( F |` R ) = ( `' F |` P ) )
10		f1oeq1 5607	. . 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 1398   `'ccnv 4753   ran crn 4755   |` cres 4756   "cima 4757   Fun wfun 5351   -1-1->wf1 5354   -1-1-onto->wf1o 5356

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364

This theorem is referenced by:  f1oresrab  5847