Theorem funcocnv2 5151

Description: Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
funcocnv2	|- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) )

Proof of Theorem funcocnv2

Step	Hyp	Ref	Expression
1		df-fun 4904	. . 3 |- ( Fun F <-> ( Rel F /\ ( F o. `' F ) C_ _I ) )
2	1	simprbi 260	. 2 |- ( Fun F -> ( F o. `' F ) C_ _I )
3		iss 4654	. . 3 |- ( ( F o. `' F ) C_ _I <-> ( F o. `' F ) = ( _I |` dom ( F o. `' F ) ) )
4		dfdm4 4527	. . . . . . . 8 |- dom F = ran `' F
5		dmcoeq 4604	. . . . . . . 8 |- ( dom F = ran `' F -> dom ( F o. `' F ) = dom `' F )
6	4, 5	ax-mp 7	. . . . . . 7 |- dom ( F o. `' F ) = dom `' F
7		df-rn 4356	. . . . . . 7 |- ran F = dom `' F
8	6, 7	eqtr4i 2063	. . . . . 6 |- dom ( F o. `' F ) = ran F
9	8	a1i 9	. . . . 5 |- ( Fun F -> dom ( F o. `' F ) = ran F )
10	9	reseq2d 4612	. . . 4 |- ( Fun F -> ( _I |` dom ( F o. `' F ) ) = ( _I |` ran F ) )
11	10	eqeq2d 2051	. . 3 |- ( Fun F -> ( ( F o. `' F ) = ( _I |` dom ( F o. `' F ) ) <-> ( F o. `' F ) = ( _I |` ran F ) ) )
12	3, 11	syl5bb 181	. 2 |- ( Fun F -> ( ( F o. `' F ) C_ _I <-> ( F o. `' F ) = ( _I |` ran F ) ) )
13	2, 12	mpbid 135	1 |- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) )

Syntax hints:   -> wi 4   = wceq 1243   C_ wss 2917   _I cid 4025   `'ccnv 4344   dom cdm 4345   ran crn 4346   |` cres 4347   o. ccom 4349   Rel wrel 4350   Fun wfun 4896

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-fun 4904

This theorem is referenced by:  fococnv2  5152  f1cocnv2  5154  funcoeqres  5157