Theorem cocnv 15716

Description: Composition with a function and then with the converse.

Assertion

Ref	Expression
cocnv	|- ((Fun F /\ Fun G) -> ((F o. G) o. `'G) = (F |` ran G))

Proof of Theorem cocnv

Step	Hyp	Ref	Expression
1		cnvcan 15715	. . . . 5 |- (Fun G -> (G o. `'G) = ( _I |` ran G))
2	1	adantl 424	. . . 4 |- ((Fun F /\ Fun G) -> (G o. `'G) = ( _I |` ran G))
3	2	coeq2d 4128	. . 3 |- ((Fun F /\ Fun G) -> (F o. (G o. `'G)) = (F o. ( _I |` ran G)))
4		funrel 4438	. . . . . 6 |- (Fun F -> Rel F)
5		coi1 4413	. . . . . 6 |- (Rel F -> (F o. _I ) = F)
6		reseq1 4218	. . . . . 6 |- ((F o. _I ) = F -> ((F o. _I ) |` ran G) = (F |` ran G))
7	4, 5, 6	3syl 24	. . . . 5 |- (Fun F -> ((F o. _I ) |` ran G) = (F |` ran G))
8	7	adantr 425	. . . 4 |- ((Fun F /\ Fun G) -> ((F o. _I ) |` ran G) = (F |` ran G))
9		resco 4402	. . . 4 |- ((F o. _I ) |` ran G) = (F o. ( _I |` ran G))
10	8, 9	syl5eqr 1942	. . 3 |- ((Fun F /\ Fun G) -> (F o. ( _I |` ran G)) = (F |` ran G))
11	3, 10	eqtrd 1925	. 2 |- ((Fun F /\ Fun G) -> (F o. (G o. `'G)) = (F |` ran G))
12		coass 4415	. 2 |- ((F o. G) o. `'G) = (F o. (G o. `'G))
13	11, 12	syl5eq 1940	1 |- ((Fun F /\ Fun G) -> ((F o. G) o. `'G) = (F |` ran G))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   _I cid 3582  `'ccnv 3985  ran crn 3987   |` cres 3988   o. ccom 3990  Rel wrel 3991  Fun wfun 3992

This theorem is referenced by:  f1ocan1fv 15717

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-fun 4008

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