Theorem funcocnv2 5467

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 5200	. . 3 |- ( Fun F <-> ( Rel F /\ ( F o. `' F ) C_ _I ) )
2	1	simprbi 273	. 2 |- ( Fun F -> ( F o. `' F ) C_ _I )
3		iss 4937	. . 3 |- ( ( F o. `' F ) C_ _I <-> ( F o. `' F ) = ( _I |` dom ( F o. `' F ) ) )
4		dfdm4 4803	. . . . . . . 8 |- dom F = ran `' F
5		dmcoeq 4883	. . . . . . . 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 4622	. . . . . . 7 |- ran F = dom `' F
8	6, 7	eqtr4i 2194	. . . . . 6 |- dom ( F o. `' F ) = ran F
9	8	a1i 9	. . . . 5 |- ( Fun F -> dom ( F o. `' F ) = ran F )
10	9	reseq2d 4891	. . . 4 |- ( Fun F -> ( _I |` dom ( F o. `' F ) ) = ( _I |` ran F ) )
11	10	eqeq2d 2182	. . 3 |- ( Fun F -> ( ( F o. `' F ) = ( _I |` dom ( F o. `' F ) ) <-> ( F o. `' F ) = ( _I |` ran F ) ) )
12	3, 11	syl5bb 191	. 2 |- ( Fun F -> ( ( F o. `' F ) C_ _I <-> ( F o. `' F ) = ( _I |` ran F ) ) )
13	2, 12	mpbid 146	1 |- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) )

Syntax hints:   -> wi 4   = wceq 1348   C_ wss 3121   _I cid 4273   `'ccnv 4610   dom cdm 4611   ran crn 4612   |` cres 4613   o. ccom 4615   Rel wrel 4616   Fun wfun 5192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-fun 5200

This theorem is referenced by:  fococnv2  5468  f1cocnv2  5470  funcoeqres  5473