Theorem ofco2 22472

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 1193	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → Fun 𝐻)
2		fvimacnvi 7071	. . . 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 6598	. . . . . 6 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → 𝐻 Fn dom 𝐻)
5		dffn5 6966	. . . . . 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 5998	. . . 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 6101	. . . . 5 ⊢ (◡𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ⊆ dom 𝐻
9		resmpt 6056	. . . . 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 2790	. . 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 8005	. . . 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 6906	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐻‘𝑥)))
15		fveq2 6906	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐺‘𝑦) = (𝐺‘(𝐻‘𝑥)))
16	14, 15	oveq12d 7448	. . 3 ⊢ (𝑦 = (𝐻‘𝑥) → ((𝐹‘𝑦)𝑅(𝐺‘𝑦)) = ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥))))
17	3, 11, 13, 16	fmptco 7148	. 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 7463	. . . . . . . 8 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V
19	18	rgenw 3062	. . . . . . 7 ⊢ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)((𝐹‘𝑥)𝑅(𝐺‘𝑥)) ∈ V
20		eqid 2734	. . . . . . . 8 ⊢ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
21	20	fnmpt 6708	. . . . . . 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 8005	. . . . . . . 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 6661	. . . . . 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 6673	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → dom (𝐹 ∘f 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺))
28		eqimss 4053	. . . 4 ⊢ (dom (𝐹 ∘f 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺) → dom (𝐹 ∘f 𝑅𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺))
29		cores2 6280	. . . 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 6647	. . . . 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 5875	. . 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 2776	. 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 1194	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝐹 ∘ 𝐻) ∈ V)
36		simpr3 1195	. . . 4 ⊢ (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) → (𝐺 ∘ 𝐻) ∈ V)
37		offval3 8005	. . . 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 6275	. . . . . 6 ⊢ dom (𝐹 ∘ 𝐻) = (◡𝐻 “ dom 𝐹)
40		dmco 6275	. . . . . 6 ⊢ dom (𝐺 ∘ 𝐻) = (◡𝐻 “ dom 𝐺)
41	39, 40	ineq12i 4225	. . . . 5 ⊢ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) = ((◡𝐻 “ dom 𝐹) ∩ (◡𝐻 “ dom 𝐺))
42		inpreima 7083	. . . . . 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 2793	. . . 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 1214	. . . . . 6 ⊢ ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹 ∘ 𝐻) ∈ V ∧ (𝐺 ∘ 𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻))) → Fun 𝐻)
46		inss2 4245	. . . . . . . . 9 ⊢ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ⊆ dom (𝐺 ∘ 𝐻)
47		dmcoss 5987	. . . . . . . . 9 ⊢ dom (𝐺 ∘ 𝐻) ⊆ dom 𝐻
48	46, 47	sstri 4004	. . . . . . . 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 3994	. . . . . 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 7006	. . . . . 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 4244	. . . . . . . . 9 ⊢ (dom (𝐹 ∘ 𝐻) ∩ dom (𝐺 ∘ 𝐻)) ⊆ dom (𝐹 ∘ 𝐻)
54		dmcoss 5987	. . . . . . . . 9 ⊢ dom (𝐹 ∘ 𝐻) ⊆ dom 𝐻
55	53, 54	sstri 4004	. . . . . . . 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 3994	. . . . . 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 7006	. . . . . 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 7448	. . . 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 5236	. . 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 2774	. 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 2784	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 1536   ∈ wcel 2105  ∀wral 3058  Vcvv 3477   ∩ cin 3961   ⊆ wss 3962   ↦ cmpt 5230  ^◡ccnv 5687  dom cdm 5688   ↾ cres 5690   “ cima 5691   ∘ ccom 5692  Fun wfun 6556   Fn wfn 6557  ‘cfv 6562  (class class class)co 7430   ∘_f cof 7694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696

This theorem is referenced by:  oftpos  22473