Theorem funcocnv2 5532

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 5261	. . 3 |- ( Fun F <-> ( Rel F /\ ( F o. `' F ) C_ _I ) )
2	1	simprbi 275	. 2 |- ( Fun F -> ( F o. `' F ) C_ _I )
3		iss 4993	. . 3 |- ( ( F o. `' F ) C_ _I <-> ( F o. `' F ) = ( _I |` dom ( F o. `' F ) ) )
4		dfdm4 4859	. . . . . . . 8 |- dom F = ran `' F
5		dmcoeq 4939	. . . . . . . 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 4675	. . . . . . 7 |- ran F = dom `' F
8	6, 7	eqtr4i 2220	. . . . . 6 |- dom ( F o. `' F ) = ran F
9	8	a1i 9	. . . . 5 |- ( Fun F -> dom ( F o. `' F ) = ran F )
10	9	reseq2d 4947	. . . 4 |- ( Fun F -> ( _I |` dom ( F o. `' F ) ) = ( _I |` ran F ) )
11	10	eqeq2d 2208	. . 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 1364   C_ wss 3157   _I cid 4324   `'ccnv 4663   dom cdm 4664   ran crn 4665   |` cres 4666   o. ccom 4668   Rel wrel 4669   Fun wfun 5253

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-fun 5261

This theorem is referenced by:  fococnv2  5533  f1cocnv2  5535  funcoeqres  5538