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Theorem ofco2 20076
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 1060 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → Fun 𝐻)
2 fvimacnvi 6239 . . . 4 ((Fun 𝐻𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) → (𝐻𝑥) ∈ (dom 𝐹 ∩ dom 𝐺))
31, 2sylan 487 . . 3 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) → (𝐻𝑥) ∈ (dom 𝐹 ∩ dom 𝐺))
4 funfn 5833 . . . . . . 7 (Fun 𝐻𝐻 Fn dom 𝐻)
51, 4sylib 207 . . . . . 6 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → 𝐻 Fn dom 𝐻)
6 dffn5 6151 . . . . . 6 (𝐻 Fn dom 𝐻𝐻 = (𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)))
75, 6sylib 207 . . . . 5 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → 𝐻 = (𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)))
87reseq1d 5316 . . . 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 5404 . . . . 5 (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ⊆ dom 𝐻
10 resmpt 5369 . . . . 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 2660 . . 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 7053 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹𝑓 𝑅𝐺) = (𝑦 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
1413adantr 480 . . 3 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹𝑓 𝑅𝐺) = (𝑦 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
15 fveq2 6103 . . . 4 (𝑦 = (𝐻𝑥) → (𝐹𝑦) = (𝐹‘(𝐻𝑥)))
16 fveq2 6103 . . . 4 (𝑦 = (𝐻𝑥) → (𝐺𝑦) = (𝐺‘(𝐻𝑥)))
1715, 16oveq12d 6567 . . 3 (𝑦 = (𝐻𝑥) → ((𝐹𝑦)𝑅(𝐺𝑦)) = ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥))))
183, 12, 14, 17fmptco 6303 . 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 6577 . . . . . . . 8 ((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V
2019rgenw 2908 . . . . . . 7 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V
21 eqid 2610 . . . . . . . 8 (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
2221fnmpt 5933 . . . . . . 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 7053 . . . . . . . 8 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
2524adantr 480 . . . . . . 7 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
2625fneq1d 5895 . . . . . 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 246 . . . . 5 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹𝑓 𝑅𝐺) Fn (dom 𝐹 ∩ dom 𝐺))
28 fndm 5904 . . . . 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 3620 . . . 4 (dom (𝐹𝑓 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺) → dom (𝐹𝑓 𝑅𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺))
31 cores2 5565 . . . 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 5883 . . . . 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 5206 . . 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 2646 . 2 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝑓 𝑅𝐺) ∘ (𝐻 ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)))))
37 simpr2 1061 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹𝐻) ∈ V)
38 simpr3 1062 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐺𝐻) ∈ V)
39 offval3 7053 . . . 4 (((𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V) → ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)) = (𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ↦ (((𝐹𝐻)‘𝑥)𝑅((𝐺𝐻)‘𝑥))))
4037, 38, 39syl2anc 691 . . 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 6250 . . . . . 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 5560 . . . . . 6 dom (𝐹𝐻) = (𝐻 “ dom 𝐹)
44 dmco 5560 . . . . . 6 dom (𝐺𝐻) = (𝐻 “ dom 𝐺)
4543, 44ineq12i 3774 . . . . 5 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) = ((𝐻 “ dom 𝐹) ∩ (𝐻 “ dom 𝐺))
4642, 45syl6reqr 2663 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) = (𝐻 “ (dom 𝐹 ∩ dom 𝐺)))
47 simplr1 1096 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → Fun 𝐻)
48 inss2 3796 . . . . . . . . 9 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom (𝐺𝐻)
49 dmcoss 5306 . . . . . . . . 9 dom (𝐺𝐻) ⊆ dom 𝐻
5048, 49sstri 3577 . . . . . . . 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 3568 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → 𝑥 ∈ dom 𝐻)
53 fvco 6184 . . . . . 6 ((Fun 𝐻𝑥 ∈ dom 𝐻) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
5447, 52, 53syl2anc 691 . . . . 5 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
55 inss1 3795 . . . . . . . . 9 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom (𝐹𝐻)
56 dmcoss 5306 . . . . . . . . 9 dom (𝐹𝐻) ⊆ dom 𝐻
5755, 56sstri 3577 . . . . . . . 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 3568 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → 𝑥 ∈ dom 𝐻)
60 fvco 6184 . . . . . 6 ((Fun 𝐻𝑥 ∈ dom 𝐻) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
6147, 59, 60syl2anc 691 . . . . 5 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
6254, 61oveq12d 6567 . . . 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 4662 . . 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 2644 . 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 2654 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 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  cin 3539  wss 3540  cmpt 4643  ccnv 5037  dom cdm 5038  cres 5040  cima 5041  ccom 5042  Fun wfun 5798   Fn wfn 5799  cfv 5804  (class class class)co 6549  𝑓 cof 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795
This theorem is referenced by:  oftpos  20077
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