Theorem funcocnv2 5644

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 5359	. . 3 |- ( Fun F <-> ( Rel F /\ ( F o. `' F ) C_ _I ) )
2	1	simprbi 275	. 2 |- ( Fun F -> ( F o. `' F ) C_ _I )
3		iss 5089	. . 3 |- ( ( F o. `' F ) C_ _I <-> ( F o. `' F ) = ( _I |` dom ( F o. `' F ) ) )
4		dfdm4 4953	. . . . . . . 8 |- dom F = ran `' F
5		dmcoeq 5035	. . . . . . . 8 |- ( dom F = ran `' F -> dom ( F o. `' F ) = dom `' F )
6	4, 5	ax-mp 5	. . . . . . 7 |- dom ( F o. `' F ) = dom `' F
7		df-rn 4765	. . . . . . 7 |- ran F = dom `' F
8	6, 7	eqtr4i 2258	. . . . . 6 |- dom ( F o. `' F ) = ran F
9	8	a1i 9	. . . . 5 |- ( Fun F -> dom ( F o. `' F ) = ran F )
10	9	reseq2d 5043	. . . 4 |- ( Fun F -> ( _I |` dom ( F o. `' F ) ) = ( _I |` ran F ) )
11	10	eqeq2d 2246	. . 3 |- ( Fun F -> ( ( F o. `' F ) = ( _I |` dom ( F o. `' F ) ) <-> ( F o. `' F ) = ( _I |` ran F ) ) )
12	3, 11	bitrid 192	. 2 |- ( Fun F -> ( ( F o. `' F ) C_ _I <-> ( F o. `' F ) = ( _I |` ran F ) ) )
13	2, 12	mpbid 147	1 |- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3214   _I cid 4414   `'ccnv 4753   dom cdm 4754   ran crn 4755   |` cres 4756   o. ccom 4758   Rel wrel 4759   Fun wfun 5351

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-fun 5359

This theorem is referenced by:  fococnv2  5645  f1cocnv2  5647  funcoeqres  5650