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Theorem ofco2 20988
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 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹f 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘f 𝑅(𝐺𝐻)))

Proof of Theorem ofco2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr1 1186 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → Fun 𝐻)
2 fvimacnvi 6814 . . . 4 ((Fun 𝐻𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) → (𝐻𝑥) ∈ (dom 𝐹 ∩ dom 𝐺))
31, 2sylan 580 . . 3 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) → (𝐻𝑥) ∈ (dom 𝐹 ∩ dom 𝐺))
41funfnd 6379 . . . . . 6 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → 𝐻 Fn dom 𝐻)
5 dffn5 6717 . . . . . 6 (𝐻 Fn dom 𝐻𝐻 = (𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)))
64, 5sylib 219 . . . . 5 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → 𝐻 = (𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)))
76reseq1d 5845 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐻 ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) = ((𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)) ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))))
8 cnvimass 5942 . . . . 5 (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ⊆ dom 𝐻
9 resmpt 5898 . . . . 5 ((𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ⊆ dom 𝐻 → ((𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)) ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) = (𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ (𝐻𝑥)))
108, 9ax-mp 5 . . . 4 ((𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)) ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) = (𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ (𝐻𝑥))
117, 10syl6eq 2869 . . 3 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐻 ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) = (𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ (𝐻𝑥)))
12 offval3 7672 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹f 𝑅𝐺) = (𝑦 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
1312adantr 481 . . 3 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹f 𝑅𝐺) = (𝑦 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
14 fveq2 6663 . . . 4 (𝑦 = (𝐻𝑥) → (𝐹𝑦) = (𝐹‘(𝐻𝑥)))
15 fveq2 6663 . . . 4 (𝑦 = (𝐻𝑥) → (𝐺𝑦) = (𝐺‘(𝐻𝑥)))
1614, 15oveq12d 7163 . . 3 (𝑦 = (𝐻𝑥) → ((𝐹𝑦)𝑅(𝐺𝑦)) = ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥))))
173, 11, 13, 16fmptco 6883 . 2 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹f 𝑅𝐺) ∘ (𝐻 ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)))) = (𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥)))))
18 ovex 7178 . . . . . . . 8 ((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V
1918rgenw 3147 . . . . . . 7 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V
20 eqid 2818 . . . . . . . 8 (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
2120fnmpt 6481 . . . . . . 7 (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) Fn (dom 𝐹 ∩ dom 𝐺))
2219, 21mp1i 13 . . . . . 6 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) Fn (dom 𝐹 ∩ dom 𝐺))
23 offval3 7672 . . . . . . . 8 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
2423adantr 481 . . . . . . 7 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
2524fneq1d 6439 . . . . . 6 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹f 𝑅𝐺) Fn (dom 𝐹 ∩ dom 𝐺) ↔ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) Fn (dom 𝐹 ∩ dom 𝐺)))
2622, 25mpbird 258 . . . . 5 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹f 𝑅𝐺) Fn (dom 𝐹 ∩ dom 𝐺))
27 fndm 6448 . . . . 5 ((𝐹f 𝑅𝐺) Fn (dom 𝐹 ∩ dom 𝐺) → dom (𝐹f 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺))
2826, 27syl 17 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → dom (𝐹f 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺))
29 eqimss 4020 . . . 4 (dom (𝐹f 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺) → dom (𝐹f 𝑅𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺))
30 cores2 6105 . . . 4 (dom (𝐹f 𝑅𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺) → ((𝐹f 𝑅𝐺) ∘ (𝐻 ↾ (dom 𝐹 ∩ dom 𝐺))) = ((𝐹f 𝑅𝐺) ∘ 𝐻))
3128, 29, 303syl 18 . . 3 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹f 𝑅𝐺) ∘ (𝐻 ↾ (dom 𝐹 ∩ dom 𝐺))) = ((𝐹f 𝑅𝐺) ∘ 𝐻))
32 funcnvres2 6427 . . . . 5 (Fun 𝐻(𝐻 ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝐻 ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))))
331, 32syl 17 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐻 ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝐻 ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))))
3433coeq2d 5726 . . 3 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹f 𝑅𝐺) ∘ (𝐻 ↾ (dom 𝐹 ∩ dom 𝐺))) = ((𝐹f 𝑅𝐺) ∘ (𝐻 ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)))))
3531, 34eqtr3d 2855 . 2 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹f 𝑅𝐺) ∘ 𝐻) = ((𝐹f 𝑅𝐺) ∘ (𝐻 ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)))))
36 simpr2 1187 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹𝐻) ∈ V)
37 simpr3 1188 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐺𝐻) ∈ V)
38 offval3 7672 . . . 4 (((𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V) → ((𝐹𝐻) ∘f 𝑅(𝐺𝐻)) = (𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ↦ (((𝐹𝐻)‘𝑥)𝑅((𝐺𝐻)‘𝑥))))
3936, 37, 38syl2anc 584 . . 3 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹𝐻) ∘f 𝑅(𝐺𝐻)) = (𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ↦ (((𝐹𝐻)‘𝑥)𝑅((𝐺𝐻)‘𝑥))))
40 inpreima 6826 . . . . . 6 (Fun 𝐻 → (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) = ((𝐻 “ dom 𝐹) ∩ (𝐻 “ dom 𝐺)))
411, 40syl 17 . . . . 5 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) = ((𝐻 “ dom 𝐹) ∩ (𝐻 “ dom 𝐺)))
42 dmco 6100 . . . . . 6 dom (𝐹𝐻) = (𝐻 “ dom 𝐹)
43 dmco 6100 . . . . . 6 dom (𝐺𝐻) = (𝐻 “ dom 𝐺)
4442, 43ineq12i 4184 . . . . 5 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) = ((𝐻 “ dom 𝐹) ∩ (𝐻 “ dom 𝐺))
4541, 44syl6reqr 2872 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) = (𝐻 “ (dom 𝐹 ∩ dom 𝐺)))
46 simplr1 1207 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → Fun 𝐻)
47 inss2 4203 . . . . . . . . 9 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom (𝐺𝐻)
48 dmcoss 5835 . . . . . . . . 9 dom (𝐺𝐻) ⊆ dom 𝐻
4947, 48sstri 3973 . . . . . . . 8 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom 𝐻
5049a1i 11 . . . . . . 7 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom 𝐻)
5150sselda 3964 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → 𝑥 ∈ dom 𝐻)
52 fvco 6752 . . . . . 6 ((Fun 𝐻𝑥 ∈ dom 𝐻) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
5346, 51, 52syl2anc 584 . . . . 5 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
54 inss1 4202 . . . . . . . . 9 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom (𝐹𝐻)
55 dmcoss 5835 . . . . . . . . 9 dom (𝐹𝐻) ⊆ dom 𝐻
5654, 55sstri 3973 . . . . . . . 8 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom 𝐻
5756a1i 11 . . . . . . 7 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom 𝐻)
5857sselda 3964 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → 𝑥 ∈ dom 𝐻)
59 fvco 6752 . . . . . 6 ((Fun 𝐻𝑥 ∈ dom 𝐻) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
6046, 58, 59syl2anc 584 . . . . 5 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
6153, 60oveq12d 7163 . . . 4 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → (((𝐹𝐻)‘𝑥)𝑅((𝐺𝐻)‘𝑥)) = ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥))))
6245, 61mpteq12dva 5141 . . 3 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ↦ (((𝐹𝐻)‘𝑥)𝑅((𝐺𝐻)‘𝑥))) = (𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥)))))
6339, 62eqtrd 2853 . 2 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹𝐻) ∘f 𝑅(𝐺𝐻)) = (𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥)))))
6417, 35, 633eqtr4d 2863 1 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹f 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘f 𝑅(𝐺𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cin 3932  wss 3933  cmpt 5137  ccnv 5547  dom cdm 5548  cres 5550  cima 5551  ccom 5552  Fun wfun 6342   Fn wfn 6343  cfv 6348  (class class class)co 7145  f cof 7396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398
This theorem is referenced by:  oftpos  20989
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