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Theorem ofco2 21035
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 1191 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → Fun 𝐻)
2 fvimacnvi 6795 . . . 4 ((Fun 𝐻𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) → (𝐻𝑥) ∈ (dom 𝐹 ∩ dom 𝐺))
31, 2sylan 583 . . 3 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) → (𝐻𝑥) ∈ (dom 𝐹 ∩ dom 𝐺))
41funfnd 6359 . . . . . 6 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → 𝐻 Fn dom 𝐻)
5 dffn5 6697 . . . . . 6 (𝐻 Fn dom 𝐻𝐻 = (𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)))
64, 5sylib 221 . . . . 5 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → 𝐻 = (𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)))
76reseq1d 5825 . . . 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 5922 . . . . 5 (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ⊆ dom 𝐻
9 resmpt 5878 . . . . 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 2872 . . 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 7658 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹f 𝑅𝐺) = (𝑦 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
1312adantr 484 . . 3 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹f 𝑅𝐺) = (𝑦 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
14 fveq2 6643 . . . 4 (𝑦 = (𝐻𝑥) → (𝐹𝑦) = (𝐹‘(𝐻𝑥)))
15 fveq2 6643 . . . 4 (𝑦 = (𝐻𝑥) → (𝐺𝑦) = (𝐺‘(𝐻𝑥)))
1614, 15oveq12d 7148 . . 3 (𝑦 = (𝐻𝑥) → ((𝐹𝑦)𝑅(𝐺𝑦)) = ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥))))
173, 11, 13, 16fmptco 6864 . 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 7163 . . . . . . . 8 ((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V
1918rgenw 3138 . . . . . . 7 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V
20 eqid 2821 . . . . . . . 8 (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
2120fnmpt 6461 . . . . . . 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 7658 . . . . . . . 8 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
2423adantr 484 . . . . . . 7 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
2524fneq1d 6419 . . . . . 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 260 . . . . 5 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹f 𝑅𝐺) Fn (dom 𝐹 ∩ dom 𝐺))
27 fndm 6428 . . . . 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 3999 . . . 4 (dom (𝐹f 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺) → dom (𝐹f 𝑅𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺))
30 cores2 6085 . . . 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 6407 . . . . 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 5706 . . 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 2858 . 2 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹f 𝑅𝐺) ∘ 𝐻) = ((𝐹f 𝑅𝐺) ∘ (𝐻 ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)))))
36 simpr2 1192 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹𝐻) ∈ V)
37 simpr3 1193 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐺𝐻) ∈ V)
38 offval3 7658 . . . 4 (((𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V) → ((𝐹𝐻) ∘f 𝑅(𝐺𝐻)) = (𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ↦ (((𝐹𝐻)‘𝑥)𝑅((𝐺𝐻)‘𝑥))))
3936, 37, 38syl2anc 587 . . 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 6807 . . . . . 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 6080 . . . . . 6 dom (𝐹𝐻) = (𝐻 “ dom 𝐹)
43 dmco 6080 . . . . . 6 dom (𝐺𝐻) = (𝐻 “ dom 𝐺)
4442, 43ineq12i 4162 . . . . 5 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) = ((𝐻 “ dom 𝐹) ∩ (𝐻 “ dom 𝐺))
4541, 44syl6reqr 2875 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) = (𝐻 “ (dom 𝐹 ∩ dom 𝐺)))
46 simplr1 1212 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → Fun 𝐻)
47 inss2 4181 . . . . . . . . 9 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom (𝐺𝐻)
48 dmcoss 5815 . . . . . . . . 9 dom (𝐺𝐻) ⊆ dom 𝐻
4947, 48sstri 3952 . . . . . . . 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 3943 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → 𝑥 ∈ dom 𝐻)
52 fvco 6732 . . . . . 6 ((Fun 𝐻𝑥 ∈ dom 𝐻) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
5346, 51, 52syl2anc 587 . . . . 5 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
54 inss1 4180 . . . . . . . . 9 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom (𝐹𝐻)
55 dmcoss 5815 . . . . . . . . 9 dom (𝐹𝐻) ⊆ dom 𝐻
5654, 55sstri 3952 . . . . . . . 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 3943 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → 𝑥 ∈ dom 𝐻)
59 fvco 6732 . . . . . 6 ((Fun 𝐻𝑥 ∈ dom 𝐻) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
6046, 58, 59syl2anc 587 . . . . 5 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
6153, 60oveq12d 7148 . . . 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 5123 . . 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 2856 . 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 2866 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 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  Vcvv 3471  cin 3909  wss 3910  cmpt 5119  ccnv 5527  dom cdm 5528  cres 5530  cima 5531  ccom 5532  Fun wfun 6322   Fn wfn 6323  cfv 6328  (class class class)co 7130  f cof 7382
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-of 7384
This theorem is referenced by:  oftpos  21036
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