Theorem fvco2 2866

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 ∈ A) → ((F ∘ G) ‘C) = (F ‘(G ‘C)))

Proof of Theorem fvco2

Step	Hyp	Ref	Expression
1		fvco 2865	. . . . . . . 8 ⊢ (((Fun F ∧ Fun G) ∧ C ∈ dom G) → ((F ∘ G) ‘C) = (F ‘(G ‘C)))
2	1	exp31 293	. . . . . . 7 ⊢ (Fun F → (Fun G → (C ∈ dom G → ((F ∘ G) ‘C) = (F ‘(G ‘C)))))
3	2	com3l 34	. . . . . 6 ⊢ (Fun G → (C ∈ dom G → (Fun F → ((F ∘ G) ‘C) = (F ‘(G ‘C)))))
4	3	imp 277	. . . . 5 ⊢ ((Fun G ∧ C ∈ dom G) → (Fun F → ((F ∘ G) ‘C) = (F ‘(G ‘C))))
5	4	funfni 2724	. . . 4 ⊢ ((G Fn A ∧ C ∈ A) → (Fun F → ((F ∘ G) ‘C) = (F ‘(G ‘C))))
6	5	exp 291	. . 3 ⊢ (G Fn A → (C ∈ A → (Fun F → ((F ∘ G) ‘C) = (F ‘(G ‘C)))))
7	6	com3r 35	. 2 ⊢ (Fun F → (G Fn A → (C ∈ A → ((F ∘ G) ‘C) = (F ‘(G ‘C)))))
8	7	imp31 280	1 ⊢ (((Fun F ∧ G Fn A) ∧ C ∈ A) → ((F ∘ G) ‘C) = (F ‘(G ‘C)))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  dom cdm 2410   ∘ ccom 2414  Fun wfun 2416   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  fvco3 2867  ruclem10 4894  ruclem11 4895

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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