Theorem funcoeqres 5406

Description: Express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)

Assertion

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

Proof of Theorem funcoeqres

Step	Hyp	Ref	Expression
1		funcocnv2 5400	. . . 4 |- ( Fun G -> ( G o. `' G ) = ( _I |` ran G ) )
2	1	coeq2d 4709	. . 3 |- ( Fun G -> ( F o. ( G o. `' G ) ) = ( F o. ( _I |` ran G ) ) )
3		coass 5065	. . . 4 |- ( ( F o. G ) o. `' G ) = ( F o. ( G o. `' G ) )
4	3	eqcomi 2144	. . 3 |- ( F o. ( G o. `' G ) ) = ( ( F o. G ) o. `' G )
5		coires1 5064	. . 3 |- ( F o. ( _I |` ran G ) ) = ( F |` ran G )
6	2, 4, 5	3eqtr3g 2196	. 2 |- ( Fun G -> ( ( F o. G ) o. `' G ) = ( F |` ran G ) )
7		coeq1 4704	. 2 |- ( ( F o. G ) = H -> ( ( F o. G ) o. `' G ) = ( H o. `' G ) )
8	6, 7	sylan9req 2194	1 |- ( ( Fun G /\ ( F o. G ) = H ) -> ( F |` ran G ) = ( H o. `' G ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   _I cid 4218   `'ccnv 4546   ran crn 4548   |` cres 4549   o. ccom 4551   Fun wfun 5125

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-fun 5133

This theorem is referenced by: (None)