Theorem funcocnv2 5182

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 4934	. . 3 |- ( Fun F <-> ( Rel F /\ ( F o. `' F ) C_ _I ) )
2	1	simprbi 269	. 2 |- ( Fun F -> ( F o. `' F ) C_ _I )
3		iss 4684	. . 3 |- ( ( F o. `' F ) C_ _I <-> ( F o. `' F ) = ( _I |` dom ( F o. `' F ) ) )
4		dfdm4 4555	. . . . . . . 8 |- dom F = ran `' F
5		dmcoeq 4632	. . . . . . . 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 4382	. . . . . . 7 |- ran F = dom `' F
8	6, 7	eqtr4i 2105	. . . . . 6 |- dom ( F o. `' F ) = ran F
9	8	a1i 9	. . . . 5 |- ( Fun F -> dom ( F o. `' F ) = ran F )
10	9	reseq2d 4640	. . . 4 |- ( Fun F -> ( _I |` dom ( F o. `' F ) ) = ( _I |` ran F ) )
11	10	eqeq2d 2093	. . 3 |- ( Fun F -> ( ( F o. `' F ) = ( _I |` dom ( F o. `' F ) ) <-> ( F o. `' F ) = ( _I |` ran F ) ) )
12	3, 11	syl5bb 190	. 2 |- ( Fun F -> ( ( F o. `' F ) C_ _I <-> ( F o. `' F ) = ( _I |` ran F ) ) )
13	2, 12	mpbid 145	1 |- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) )

Syntax hints:   -> wi 4   = wceq 1285   C_ wss 2974   _I cid 4051   `'ccnv 4370   dom cdm 4371   ran crn 4372   |` cres 4373   o. ccom 4375   Rel wrel 4376   Fun wfun 4926

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-fun 4934

This theorem is referenced by:  fococnv2  5183  f1cocnv2  5185  funcoeqres  5188