Theorem ofco2 21056

Description: Distribution law for the function operation and the composition of functions. (Contributed by Stefan O'Rear, 17-Jul-2018.)

Assertion

Ref	Expression
ofco2	⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)))

Proof of Theorem ofco2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr1 1191	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → Fun 𝐻)
2		fvimacnvi 6799	. . . 4 ⊢ ((Fun 𝐻 ∧ 𝑥 ∈ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺))) → (𝐻‘𝑥) ∈ (dom 𝐹 ∩ dom 𝐺))
3	1, 2	sylan 583	. . 3 ⊢ ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) ∧ 𝑥 ∈ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺))) → (𝐻‘𝑥) ∈ (dom 𝐹 ∩ dom 𝐺))
4	1	funfnd 6355	. . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → 𝐻 Fn dom 𝐻)
5		dffn5 6699	. . . . . 6 ⊢ (𝐻 Fn dom 𝐻 ↔ 𝐻 = (𝑥 ∈ dom 𝐻 ↦ (𝐻‘𝑥)))
6	4, 5	sylib 221	. . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → 𝐻 = (𝑥 ∈ dom 𝐻 ↦ (𝐻‘𝑥)))
7	6	reseq1d 5817	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝐻 ↾ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺))) = ((𝑥 ∈ dom 𝐻 ↦ (𝐻‘𝑥)) ↾ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺))))
8		cnvimass 5916	. . . . 5 ⊢ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ⊆ dom 𝐻
9		resmpt 5872	. . . . 5 ⊢ ((◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ⊆ dom 𝐻 → ((𝑥 ∈ dom 𝐻 ↦ (𝐻‘𝑥)) ↾ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺))) = (𝑥 ∈ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ (𝐻‘𝑥)))
10	8, 9	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ dom 𝐻 ↦ (𝐻‘𝑥)) ↾ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺))) = (𝑥 ∈ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ (𝐻‘𝑥))
11	7, 10	eqtrdi 2849	. . 3 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝐻 ↾ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺))) = (𝑥 ∈ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ (𝐻‘𝑥)))
12		offval3 7665	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘f 𝑅𝐺) = (𝑦 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑦)𝑅(𝐺‘𝑦))))
13	12	adantr 484	. . 3 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝐹 ∘f 𝑅𝐺) = (𝑦 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑦)𝑅(𝐺‘𝑦))))
14		fveq2 6645	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐻‘𝑥)))
15		fveq2 6645	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐺‘𝑦) = (𝐺‘(𝐻‘𝑥)))
16	14, 15	oveq12d 7153	. . 3 ⊢ (𝑦 = (𝐻‘𝑥) → ((𝐹‘𝑦)𝑅(𝐺‘𝑦)) = ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥))))
17	3, 11, 13, 16	fmptco 6868	. 2 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → ((𝐹 ∘f 𝑅𝐺) ∘ (𝐻 ↾ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)))) = (𝑥 ∈ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))))
18		ovex 7168	. . . . . . . 8 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V
19	18	rgenw 3118	. . . . . . 7 ⊢ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V
20		eqid 2798	. . . . . . . 8 ⊢ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
21	20	fnmpt 6460	. . . . . . 7 ⊢ (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) Fn (dom 𝐹 ∩ dom 𝐺))
22	19, 21	mp1i 13	. . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) Fn (dom 𝐹 ∩ dom 𝐺))
23		offval3 7665	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
24	23	adantr 484	. . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
25	24	fneq1d 6416	. . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → ((𝐹 ∘f 𝑅𝐺) Fn (dom 𝐹 ∩ dom 𝐺) ↔ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) Fn (dom 𝐹 ∩ dom 𝐺)))
26	22, 25	mpbird 260	. . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝐹 ∘f 𝑅𝐺) Fn (dom 𝐹 ∩ dom 𝐺))
27	26	fndmd 6427	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → dom (𝐹 ∘f 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺))
28		eqimss 3971	. . . 4 ⊢ (dom (𝐹 ∘f 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺) → dom (𝐹 ∘f 𝑅𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺))
29		cores2 6079	. . . 4 ⊢ (dom (𝐹 ∘f 𝑅𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺) → ((𝐹 ∘f 𝑅𝐺) ∘ ◡(◡𝐻 ↾ (dom 𝐹 ∩ dom 𝐺))) = ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻))
30	27, 28, 29	3syl 18	. . 3 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → ((𝐹 ∘f 𝑅𝐺) ∘ ◡(◡𝐻 ↾ (dom 𝐹 ∩ dom 𝐺))) = ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻))
31		funcnvres2 6404	. . . . 5 ⊢ (Fun 𝐻 → ◡(◡𝐻 ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝐻 ↾ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺))))
32	1, 31	syl 17	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → ◡(◡𝐻 ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝐻 ↾ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺))))
33	32	coeq2d 5697	. . 3 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → ((𝐹 ∘f 𝑅𝐺) ∘ ◡(◡𝐻 ↾ (dom 𝐹 ∩ dom 𝐺))) = ((𝐹 ∘f 𝑅𝐺) ∘ (𝐻 ↾ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)))))
34	30, 33	eqtr3d 2835	. 2 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘f 𝑅𝐺) ∘ (𝐻 ↾ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)))))
35		simpr2 1192	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝐹 ∘ 𝐻) ∈ V)
36		simpr3 1193	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝐺 ∘ 𝐻) ∈ V)
37		offval3 7665	. . . 4 ⊢ (((𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V) → ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)) = (𝑥 ∈ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ↦ (((𝐹 ∘ 𝐻)‘𝑥)𝑅((𝐺 ∘ 𝐻)‘𝑥))))
38	35, 36, 37	syl2anc 587	. . 3 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)) = (𝑥 ∈ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ↦ (((𝐹 ∘ 𝐻)‘𝑥)𝑅((𝐺 ∘ 𝐻)‘𝑥))))
39		dmco 6074	. . . . . 6 ⊢ dom (𝐹 ∘ 𝐻) = (◡𝐻 “ dom 𝐹)
40		dmco 6074	. . . . . 6 ⊢ dom (𝐺 ∘ 𝐻) = (◡𝐻 “ dom 𝐺)
41	39, 40	ineq12i 4137	. . . . 5 ⊢ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) = ((◡𝐻 “ dom 𝐹) ∩ (◡𝐻 “ dom 𝐺))
42		inpreima 6811	. . . . . 6 ⊢ (Fun 𝐻 → (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)) = ((◡𝐻 “ dom 𝐹) ∩ (◡𝐻 “ dom 𝐺)))
43	1, 42	syl 17	. . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)) = ((◡𝐻 “ dom 𝐹) ∩ (◡𝐻 “ dom 𝐺)))
44	41, 43	eqtr4id 2852	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) = (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)))
45		simplr1 1212	. . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻))) → Fun 𝐻)
46		inss2 4156	. . . . . . . . 9 ⊢ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ⊆ dom (𝐺 ∘ 𝐻)
47		dmcoss 5807	. . . . . . . . 9 ⊢ dom (𝐺 ∘ 𝐻) ⊆ dom 𝐻
48	46, 47	sstri 3924	. . . . . . . 8 ⊢ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ⊆ dom 𝐻
49	48	a1i 11	. . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ⊆ dom 𝐻)
50	49	sselda 3915	. . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻))) → 𝑥 ∈ dom 𝐻)
51		fvco 6736	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑥 ∈ dom 𝐻) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
52	45, 50, 51	syl2anc 587	. . . . 5 ⊢ ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻))) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
53		inss1 4155	. . . . . . . . 9 ⊢ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ⊆ dom (𝐹 ∘ 𝐻)
54		dmcoss 5807	. . . . . . . . 9 ⊢ dom (𝐹 ∘ 𝐻) ⊆ dom 𝐻
55	53, 54	sstri 3924	. . . . . . . 8 ⊢ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ⊆ dom 𝐻
56	55	a1i 11	. . . . . . 7 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ⊆ dom 𝐻)
57	56	sselda 3915	. . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻))) → 𝑥 ∈ dom 𝐻)
58		fvco 6736	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑥 ∈ dom 𝐻) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
59	45, 57, 58	syl2anc 587	. . . . 5 ⊢ ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻))) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
60	52, 59	oveq12d 7153	. . . 4 ⊢ ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻))) → (((𝐹 ∘ 𝐻)‘𝑥)𝑅((𝐺 ∘ 𝐻)‘𝑥)) = ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥))))
61	44, 60	mpteq12dva 5114	. . 3 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝑥 ∈ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ↦ (((𝐹 ∘ 𝐻)‘𝑥)𝑅((𝐺 ∘ 𝐻)‘𝑥))) = (𝑥 ∈ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))))
62	38, 61	eqtrd 2833	. 2 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)) = (𝑥 ∈ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))))
63	17, 34, 62	3eqtr4d 2843	1 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881   ↦ cmpt 5110  ^◡ccnv 5518  dom cdm 5519   ↾ cres 5521   “ cima 5522   ∘ ccom 5523  Fun wfun 6318   Fn wfn 6319  ‘cfv 6324  (class class class)co 7135   ∘_f cof 7387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389

This theorem is referenced by:  oftpos  21057