Theorem funcoeqres 5665

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 5659	. . . 4 |- ( Fun G -> ( G o. `' G ) = ( _I |` ran G ) )
2	1	coeq2d 4937	. . 3 |- ( Fun G -> ( F o. ( G o. `' G ) ) = ( F o. ( _I |` ran G ) ) )
3		coass 5301	. . . 4 |- ( ( F o. G ) o. `' G ) = ( F o. ( G o. `' G ) )
4	3	eqcomi 2242	. . 3 |- ( F o. ( G o. `' G ) ) = ( ( F o. G ) o. `' G )
5		coires1 5300	. . 3 |- ( F o. ( _I |` ran G ) ) = ( F |` ran G )
6	2, 4, 5	3eqtr3g 2294	. 2 |- ( Fun G -> ( ( F o. G ) o. `' G ) = ( F |` ran G ) )
7		coeq1 4932	. 2 |- ( ( F o. G ) = H -> ( ( F o. G ) o. `' G ) = ( H o. `' G ) )
8	6, 7	sylan9req 2292	1 |- ( ( Fun G /\ ( F o. G ) = H ) -> ( F |` ran G ) = ( H o. `' G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   _I cid 4428   `'ccnv 4768   ran crn 4770   |` cres 4771   o. ccom 4773   Fun wfun 5366

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-fun 5374

This theorem is referenced by: (None)