Theorem funcoeqres 5494

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 5488	. . . 4 |- ( Fun G -> ( G o. `' G ) = ( _I |` ran G ) )
2	1	coeq2d 4791	. . 3 |- ( Fun G -> ( F o. ( G o. `' G ) ) = ( F o. ( _I |` ran G ) ) )
3		coass 5149	. . . 4 |- ( ( F o. G ) o. `' G ) = ( F o. ( G o. `' G ) )
4	3	eqcomi 2181	. . 3 |- ( F o. ( G o. `' G ) ) = ( ( F o. G ) o. `' G )
5		coires1 5148	. . 3 |- ( F o. ( _I |` ran G ) ) = ( F |` ran G )
6	2, 4, 5	3eqtr3g 2233	. 2 |- ( Fun G -> ( ( F o. G ) o. `' G ) = ( F |` ran G ) )
7		coeq1 4786	. 2 |- ( ( F o. G ) = H -> ( ( F o. G ) o. `' G ) = ( H o. `' G ) )
8	6, 7	sylan9req 2231	1 |- ( ( Fun G /\ ( F o. G ) = H ) -> ( F |` ran G ) = ( H o. `' G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   _I cid 4290   `'ccnv 4627   ran crn 4629   |` cres 4630   o. ccom 4632   Fun wfun 5212

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-fun 5220

This theorem is referenced by: (None)