Theorem ofco2 22338

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 1195	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → Fun 𝐻)
2		fvimacnvi 7024	. . . 4 ⊢ ((Fun 𝐻 ∧ 𝑥 ∈ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺))) → (𝐻‘𝑥) ∈ (dom 𝐹 ∩ dom 𝐺))
3	1, 2	sylan 580	. . 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 6547	. . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → 𝐻 Fn dom 𝐻)
5		dffn5 6919	. . . . . 6 ⊢ (𝐻 Fn dom 𝐻 ↔ 𝐻 = (𝑥 ∈ dom 𝐻 ↦ (𝐻‘𝑥)))
6	4, 5	sylib 218	. . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → 𝐻 = (𝑥 ∈ dom 𝐻 ↦ (𝐻‘𝑥)))
7	6	reseq1d 5949	. . . 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 6053	. . . . 5 ⊢ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ⊆ dom 𝐻
9		resmpt 6008	. . . . 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 2780	. . 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 7961	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘f 𝑅𝐺) = (𝑦 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑦)𝑅(𝐺‘𝑦))))
13	12	adantr 480	. . 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 6858	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐻‘𝑥)))
15		fveq2 6858	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐺‘𝑦) = (𝐺‘(𝐻‘𝑥)))
16	14, 15	oveq12d 7405	. . 3 ⊢ (𝑦 = (𝐻‘𝑥) → ((𝐹‘𝑦)𝑅(𝐺‘𝑦)) = ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥))))
17	3, 11, 13, 16	fmptco 7101	. 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 7420	. . . . . . . 8 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V
19	18	rgenw 3048	. . . . . . 7 ⊢ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V
20		eqid 2729	. . . . . . . 8 ⊢ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
21	20	fnmpt 6658	. . . . . . 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 7961	. . . . . . . 8 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
24	23	adantr 480	. . . . . . 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 6611	. . . . . 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 257	. . . . 5 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝐹 ∘f 𝑅𝐺) Fn (dom 𝐹 ∩ dom 𝐺))
27	26	fndmd 6623	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → dom (𝐹 ∘f 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺))
28		eqimss 4005	. . . 4 ⊢ (dom (𝐹 ∘f 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺) → dom (𝐹 ∘f 𝑅𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺))
29		cores2 6232	. . . 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 6596	. . . . 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 5826	. . 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 2766	. 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 1196	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝐹 ∘ 𝐻) ∈ V)
36		simpr3 1197	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝐺 ∘ 𝐻) ∈ V)
37		offval3 7961	. . . 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 584	. . 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 6227	. . . . . 6 ⊢ dom (𝐹 ∘ 𝐻) = (◡𝐻 “ dom 𝐹)
40		dmco 6227	. . . . . 6 ⊢ dom (𝐺 ∘ 𝐻) = (◡𝐻 “ dom 𝐺)
41	39, 40	ineq12i 4181	. . . . 5 ⊢ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) = ((◡𝐻 “ dom 𝐹) ∩ (◡𝐻 “ dom 𝐺))
42		inpreima 7036	. . . . . 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 2783	. . . 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 1216	. . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻))) → Fun 𝐻)
46		inss2 4201	. . . . . . . . 9 ⊢ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ⊆ dom (𝐺 ∘ 𝐻)
47		dmcoss 5938	. . . . . . . . 9 ⊢ dom (𝐺 ∘ 𝐻) ⊆ dom 𝐻
48	46, 47	sstri 3956	. . . . . . . 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 3946	. . . . . 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 6959	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑥 ∈ dom 𝐻) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
52	45, 50, 51	syl2anc 584	. . . . 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 4200	. . . . . . . . 9 ⊢ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ⊆ dom (𝐹 ∘ 𝐻)
54		dmcoss 5938	. . . . . . . . 9 ⊢ dom (𝐹 ∘ 𝐻) ⊆ dom 𝐻
55	53, 54	sstri 3956	. . . . . . . 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 3946	. . . . . 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 6959	. . . . . 6 ⊢ ((Fun 𝐻 ∧ 𝑥 ∈ dom 𝐻) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
59	45, 57, 58	syl2anc 584	. . . . 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 7405	. . . 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 2764	. 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 2774	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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914   ↦ cmpt 5188  ^◡ccnv 5637  dom cdm 5638   ↾ cres 5640   “ cima 5641   ∘ ccom 5642  Fun wfun 6505   Fn wfn 6506  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653

This theorem is referenced by:  oftpos  22339