Theorem funcoeqres 6167

Description: Re-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 6161	. . . 4 |- ( Fun G -> ( G o. `' G ) = ( _I |` ran G ) )
2	1	coeq2d 5284	. . 3 |- ( Fun G -> ( F o. ( G o. `' G ) ) = ( F o. ( _I |` ran G ) ) )
3		coass 5654	. . . 4 |- ( ( F o. G ) o. `' G ) = ( F o. ( G o. `' G ) )
4	3	eqcomi 2631	. . 3 |- ( F o. ( G o. `' G ) ) = ( ( F o. G ) o. `' G )
5		coires1 5653	. . 3 |- ( F o. ( _I |` ran G ) ) = ( F |` ran G )
6	2, 4, 5	3eqtr3g 2679	. 2 |- ( Fun G -> ( ( F o. G ) o. `' G ) = ( F |` ran G ) )
7		coeq1 5279	. 2 |- ( ( F o. G ) = H -> ( ( F o. G ) o. `' G ) = ( H o. `' G ) )
8	6, 7	sylan9req 2677	1 |- ( ( Fun G /\ ( F o. G ) = H ) -> ( F |` ran G ) = ( H o. `' G ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   _I cid 5023   `'ccnv 5113   ran crn 5115   |` cres 5116   o. ccom 5118   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-fun 5890

This theorem is referenced by:  evlseu  19516  frlmup4  20140