Theorem fvco 5749

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 5384	. 2 |- ( Fun G <-> G Fn dom G )
2		fvco2 5748	. 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 1398   e. wcel 2205   dom cdm 4751   o. ccom 4755   Fun wfun 5348   Fn wfn 5349   ` cfv 5354

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362

This theorem is referenced by:  hashinfom  11145  hashennn  11147