Theorem ofco2 22778

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 1213	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → Fun 𝐻)
2		fvimacnvi 7059	. . . 4 ⊢ ((Fun 𝐻 ∧ 𝑥 ∈ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺))) → (𝐻‘𝑥) ∈ (dom 𝐹 ∩ dom 𝐺))
3	1, 2	sylan 592	. . 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 6579	. . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → 𝐻 Fn dom 𝐻)
5		dffn5 6951	. . . . . 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 5973	. . . 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 6204	. . . . 5 ⊢ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ⊆ dom 𝐻
9		resmpt 6034	. . . . 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 2814	. . 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 8001	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘f 𝑅𝐺) = (𝑦 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑦)𝑅(𝐺‘𝑦))))
13	12	adantr 486	. . 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 6893	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐻‘𝑥)))
15		fveq2 6893	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐺‘𝑦) = (𝐺‘(𝐻‘𝑥)))
16	14, 15	oveq12d 7445	. . 3 ⊢ (𝑦 = (𝐻‘𝑥) → ((𝐹‘𝑦)𝑅(𝐺‘𝑦)) = ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥))))
17	3, 11, 13, 16	fmptco 7138	. 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 7460	. . . . . . . 8 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V
19	18	rgenw 3083	. . . . . . 7 ⊢ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V
20		eqid 2763	. . . . . . . 8 ⊢ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
21	20	fnmpt 6687	. . . . . . 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 14	. . . . . 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 8001	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
24	23	adantr 486	. . . . . . 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 6640	. . . . . 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 6652	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → dom (𝐹 ∘f 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺))
28		eqimss 3970	. . . 4 ⊢ (dom (𝐹 ∘f 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺) → dom (𝐹 ∘f 𝑅𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺))
29		cores2 6267	. . . 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 19	. . 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 6628	. . . . 5 ⊢ (Fun 𝐻 → ◡(◡𝐻 ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝐻 ↾ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺))))
32	1, 31	syl 18	. . . 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 5844	. . 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 2800	. 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 1214	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝐹 ∘ 𝐻) ∈ V)
36		simpr3 1215	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝐺 ∘ 𝐻) ∈ V)
37		offval3 8001	. . . 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 596	. . 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 6262	. . . . . 6 ⊢ dom (𝐹 ∘ 𝐻) = (◡𝐻 “ dom 𝐹)
40		dmco 6262	. . . . . 6 ⊢ dom (𝐺 ∘ 𝐻) = (◡𝐻 “ dom 𝐺)
41	39, 40	ineq12i 4117	. . . . 5 ⊢ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) = ((◡𝐻 “ dom 𝐹) ∩ (◡𝐻 “ dom 𝐺))
42		inpreima 7071	. . . . . 6 ⊢ (Fun 𝐻 → (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)) = ((◡𝐻 “ dom 𝐹) ∩ (◡𝐻 “ dom 𝐺)))
43	1, 42	syl 18	. . . . 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 2817	. . . 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 1234	. . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻))) → Fun 𝐻)
46		inss2 4136	. . . . . . . . 9 ⊢ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ⊆ dom (𝐺 ∘ 𝐻)
47		dmcoss 5961	. . . . . . . . 9 ⊢ dom (𝐺 ∘ 𝐻) ⊆ dom 𝐻
48	46, 47	sstri 3921	. . . . . . . 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 3912	. . . . . 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 6991	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑥 ∈ dom 𝐻) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
52	45, 50, 51	syl2anc 596	. . . . 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 4135	. . . . . . . . 9 ⊢ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ⊆ dom (𝐹 ∘ 𝐻)
54		dmcoss 5961	. . . . . . . . 9 ⊢ dom (𝐹 ∘ 𝐻) ⊆ dom 𝐻
55	53, 54	sstri 3921	. . . . . . . 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 3912	. . . . . 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 6991	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑥 ∈ dom 𝐻) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
59	45, 57, 58	syl2anc 596	. . . . 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 7445	. . . 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 5193	. . 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 2798	. 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 2808	1 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∀wral 3079  Vcvv 3453   ⊆ wss 3899   ∩ cin 4091   ↦ cmpt 5188  ^◡ccnv 5653  dom cdm 5654   ↾ cres 5656   “ cima 5657   ∘ ccom 5658  Fun wfun 6540   Fn wfn 6541  ‘cfv 6546  (class class class)co 7427   ∘_f cof 7698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-of 7700

This theorem is used by:  oftpos  22779