Theorem funcoeqres 5535

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 5529	. . . 4 |- ( Fun G -> ( G o. `' G ) = ( _I |` ran G ) )
2	1	coeq2d 4828	. . 3 |- ( Fun G -> ( F o. ( G o. `' G ) ) = ( F o. ( _I |` ran G ) ) )
3		coass 5188	. . . 4 |- ( ( F o. G ) o. `' G ) = ( F o. ( G o. `' G ) )
4	3	eqcomi 2200	. . 3 |- ( F o. ( G o. `' G ) ) = ( ( F o. G ) o. `' G )
5		coires1 5187	. . 3 |- ( F o. ( _I |` ran G ) ) = ( F |` ran G )
6	2, 4, 5	3eqtr3g 2252	. 2 |- ( Fun G -> ( ( F o. G ) o. `' G ) = ( F |` ran G ) )
7		coeq1 4823	. 2 |- ( ( F o. G ) = H -> ( ( F o. G ) o. `' G ) = ( H o. `' G ) )
8	6, 7	sylan9req 2250	1 |- ( ( Fun G /\ ( F o. G ) = H ) -> ( F |` ran G ) = ( H o. `' G ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   _I cid 4323   `'ccnv 4662   ran crn 4664   |` cres 4665   o. ccom 4667   Fun wfun 5252

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-fun 5260

This theorem is referenced by: (None)