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Theorem ofco2 20537
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 1248 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → Fun 𝐻)
2 fvimacnvi 6523 . . . 4 ((Fun 𝐻𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) → (𝐻𝑥) ∈ (dom 𝐹 ∩ dom 𝐺))
31, 2sylan 575 . . 3 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) → (𝐻𝑥) ∈ (dom 𝐹 ∩ dom 𝐺))
4 funfn 6100 . . . . . . 7 (Fun 𝐻𝐻 Fn dom 𝐻)
51, 4sylib 209 . . . . . 6 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → 𝐻 Fn dom 𝐻)
6 dffn5 6432 . . . . . 6 (𝐻 Fn dom 𝐻𝐻 = (𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)))
75, 6sylib 209 . . . . 5 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → 𝐻 = (𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)))
87reseq1d 5566 . . . 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))))
9 cnvimass 5669 . . . . 5 (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ⊆ dom 𝐻
10 resmpt 5628 . . . . 5 ((𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ⊆ dom 𝐻 → ((𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)) ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) = (𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ (𝐻𝑥)))
119, 10ax-mp 5 . . . 4 ((𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)) ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) = (𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ (𝐻𝑥))
128, 11syl6eq 2815 . . 3 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐻 ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) = (𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ↦ (𝐻𝑥)))
13 offval3 7362 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹𝑓 𝑅𝐺) = (𝑦 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
1413adantr 472 . . 3 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹𝑓 𝑅𝐺) = (𝑦 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
15 fveq2 6377 . . . 4 (𝑦 = (𝐻𝑥) → (𝐹𝑦) = (𝐹‘(𝐻𝑥)))
16 fveq2 6377 . . . 4 (𝑦 = (𝐻𝑥) → (𝐺𝑦) = (𝐺‘(𝐻𝑥)))
1715, 16oveq12d 6862 . . 3 (𝑦 = (𝐻𝑥) → ((𝐹𝑦)𝑅(𝐺𝑦)) = ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥))))
183, 12, 14, 17fmptco 6589 . 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)))))
19 ovex 6876 . . . . . . . 8 ((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V
2019rgenw 3071 . . . . . . 7 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V
21 eqid 2765 . . . . . . . 8 (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
2221fnmpt 6200 . . . . . . 7 (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) Fn (dom 𝐹 ∩ dom 𝐺))
2320, 22mp1i 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))
24 offval3 7362 . . . . . . . 8 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
2524adantr 472 . . . . . . 7 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
2625fneq1d 6161 . . . . . 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)))
2723, 26mpbird 248 . . . . 5 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹𝑓 𝑅𝐺) Fn (dom 𝐹 ∩ dom 𝐺))
28 fndm 6170 . . . . 5 ((𝐹𝑓 𝑅𝐺) Fn (dom 𝐹 ∩ dom 𝐺) → dom (𝐹𝑓 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺))
2927, 28syl 17 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → dom (𝐹𝑓 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺))
30 eqimss 3819 . . . 4 (dom (𝐹𝑓 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺) → dom (𝐹𝑓 𝑅𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺))
31 cores2 5836 . . . 4 (dom (𝐹𝑓 𝑅𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺) → ((𝐹𝑓 𝑅𝐺) ∘ (𝐻 ↾ (dom 𝐹 ∩ dom 𝐺))) = ((𝐹𝑓 𝑅𝐺) ∘ 𝐻))
3229, 30, 313syl 18 . . 3 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹𝑓 𝑅𝐺) ∘ (𝐻 ↾ (dom 𝐹 ∩ dom 𝐺))) = ((𝐹𝑓 𝑅𝐺) ∘ 𝐻))
33 funcnvres2 6149 . . . . 5 (Fun 𝐻(𝐻 ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝐻 ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))))
341, 33syl 17 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐻 ↾ (dom 𝐹 ∩ dom 𝐺)) = (𝐻 ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))))
3534coeq2d 5455 . . 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)))))
3632, 35eqtr3d 2801 . 2 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝑓 𝑅𝐺) ∘ (𝐻 ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)))))
37 simpr2 1250 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹𝐻) ∈ V)
38 simpr3 1252 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐺𝐻) ∈ V)
39 offval3 7362 . . . 4 (((𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V) → ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)) = (𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ↦ (((𝐹𝐻)‘𝑥)𝑅((𝐺𝐻)‘𝑥))))
4037, 38, 39syl2anc 579 . . 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))))
41 inpreima 6534 . . . . . 6 (Fun 𝐻 → (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) = ((𝐻 “ dom 𝐹) ∩ (𝐻 “ dom 𝐺)))
421, 41syl 17 . . . . 5 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) = ((𝐻 “ dom 𝐹) ∩ (𝐻 “ dom 𝐺)))
43 dmco 5831 . . . . . 6 dom (𝐹𝐻) = (𝐻 “ dom 𝐹)
44 dmco 5831 . . . . . 6 dom (𝐺𝐻) = (𝐻 “ dom 𝐺)
4543, 44ineq12i 3976 . . . . 5 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) = ((𝐻 “ dom 𝐹) ∩ (𝐻 “ dom 𝐺))
4642, 45syl6reqr 2818 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) = (𝐻 “ (dom 𝐹 ∩ dom 𝐺)))
47 simplr1 1275 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → Fun 𝐻)
48 inss2 3995 . . . . . . . . 9 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom (𝐺𝐻)
49 dmcoss 5556 . . . . . . . . 9 dom (𝐺𝐻) ⊆ dom 𝐻
5048, 49sstri 3772 . . . . . . . 8 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom 𝐻
5150a1i 11 . . . . . . 7 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom 𝐻)
5251sselda 3763 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → 𝑥 ∈ dom 𝐻)
53 fvco 6465 . . . . . 6 ((Fun 𝐻𝑥 ∈ dom 𝐻) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
5447, 52, 53syl2anc 579 . . . . 5 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
55 inss1 3994 . . . . . . . . 9 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom (𝐹𝐻)
56 dmcoss 5556 . . . . . . . . 9 dom (𝐹𝐻) ⊆ dom 𝐻
5755, 56sstri 3772 . . . . . . . 8 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom 𝐻
5857a1i 11 . . . . . . 7 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom 𝐻)
5958sselda 3763 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → 𝑥 ∈ dom 𝐻)
60 fvco 6465 . . . . . 6 ((Fun 𝐻𝑥 ∈ dom 𝐻) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
6147, 59, 60syl2anc 579 . . . . 5 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
6254, 61oveq12d 6862 . . . 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))))
6346, 62mpteq12dva 4893 . . 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)))))
6440, 63eqtrd 2799 . 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)))))
6518, 36, 643eqtr4d 2809 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 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  cin 3733  wss 3734  cmpt 4890  ccnv 5278  dom cdm 5279  cres 5281  cima 5282  ccom 5283  Fun wfun 6064   Fn wfn 6065  cfv 6070  (class class class)co 6844  𝑓 cof 7095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-of 7097
This theorem is referenced by:  oftpos  20538
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