Theorem fvco2 3781

Description: Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47.

Assertion

Ref	Expression
fvco2	|- ((Fun F /\ G Fn A /\ C e. A) -> ((F o. G)` C) = (F` (G` C)))

Proof of Theorem fvco2

Step	Hyp	Ref	Expression
1		fvco 3780	. . . . . . . 8 |- ((Fun F /\ Fun G /\ C e. dom G) -> ((F o. G)` C) = (F` (G` C)))
2	1	3exp 834	. . . . . . 7 |- (Fun F -> (Fun G -> (C e. dom G -> ((F o. G)` C) = (F` (G` C)))))
3	2	com3l 34	. . . . . 6 |- (Fun G -> (C e. dom G -> (Fun F -> ((F o. G)` C) = (F` (G` C)))))
4	3	imp 350	. . . . 5 |- ((Fun G /\ C e. dom G) -> (Fun F -> ((F o. G)` C) = (F` (G` C))))
5	4	funfni 3594	. . . 4 |- ((G Fn A /\ C e. A) -> (Fun F -> ((F o. G)` C) = (F` (G` C))))
6	5	ex 373	. . 3 |- (G Fn A -> (C e. A -> (Fun F -> ((F o. G)` C) = (F` (G` C)))))
7	6	com3r 35	. 2 |- (Fun F -> (G Fn A -> (C e. A -> ((F o. G)` C) = (F` (G` C)))))
8	7	3imp 829	1 |- ((Fun F /\ G Fn A /\ C e. A) -> ((F o. G)` C) = (F` (G` C)))

Syntax hints:   -> wi 3   /\ w3a 777   = wceq 958   e. wcel 960  dom cdm 3176   o. ccom 3180  Fun wfun 3182   Fn wfn 3183  ` cfv 3188

This theorem is referenced by:  fvco3 3782  curry1 4104  ruclem10 7520  ruclem11 7521  0vfval 8221  cayleylem2 10405

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204

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