Theorem fvco 5772

Description: Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario Carneiro, 26-Dec-2014.)

Assertion

Ref	Expression
fvco	|- ( ( Fun G /\ A e. dom G ) -> ( ( F o. G ) ` A ) = ( F ` ( G ` A ) ) )

Proof of Theorem fvco

Step	Hyp	Ref	Expression
1		funfn 5405	. 2 |- ( Fun G <-> G Fn dom G )
2		fvco2 5771	. 2 |- ( ( G Fn dom G /\ A e. dom G ) -> ( ( F o. G ) ` A ) = ( F ` ( G ` A ) ) )
3	1, 2	sylanb 284	1 |- ( ( Fun G /\ A e. dom G ) -> ( ( F o. G ) ` A ) = ( F ` ( G ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   dom cdm 4772   o. ccom 4776   Fun wfun 5369   Fn wfn 5370   ` cfv 5375

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 4247  ax-pow 4309  ax-pr 4344

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-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383

This theorem is referenced by:  hashinfom  11198  hashennn  11200