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Theorem ofco2 20197
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 1065 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → Fun 𝐻)
2 fvimacnvi 6297 . . . 4 ((Fun 𝐻𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) → (𝐻𝑥) ∈ (dom 𝐹 ∩ dom 𝐺))
31, 2sylan 488 . . 3 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (𝐻 “ (dom 𝐹 ∩ dom 𝐺))) → (𝐻𝑥) ∈ (dom 𝐹 ∩ dom 𝐺))
4 funfn 5887 . . . . . . 7 (Fun 𝐻𝐻 Fn dom 𝐻)
51, 4sylib 208 . . . . . 6 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → 𝐻 Fn dom 𝐻)
6 dffn5 6208 . . . . . 6 (𝐻 Fn dom 𝐻𝐻 = (𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)))
75, 6sylib 208 . . . . 5 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → 𝐻 = (𝑥 ∈ dom 𝐻 ↦ (𝐻𝑥)))
87reseq1d 5365 . . . 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 5454 . . . . 5 (𝐻 “ (dom 𝐹 ∩ dom 𝐺)) ⊆ dom 𝐻
10 resmpt 5418 . . . . 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 2671 . . 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 7122 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹𝑓 𝑅𝐺) = (𝑦 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
1413adantr 481 . . 3 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹𝑓 𝑅𝐺) = (𝑦 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
15 fveq2 6158 . . . 4 (𝑦 = (𝐻𝑥) → (𝐹𝑦) = (𝐹‘(𝐻𝑥)))
16 fveq2 6158 . . . 4 (𝑦 = (𝐻𝑥) → (𝐺𝑦) = (𝐺‘(𝐻𝑥)))
1715, 16oveq12d 6633 . . 3 (𝑦 = (𝐻𝑥) → ((𝐹𝑦)𝑅(𝐺𝑦)) = ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥))))
183, 12, 14, 17fmptco 6362 . 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 6643 . . . . . . . 8 ((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V
2019rgenw 2920 . . . . . . 7 𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)((𝐹𝑥)𝑅(𝐺𝑥)) ∈ V
21 eqid 2621 . . . . . . . 8 (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
2221fnmpt 5987 . . . . . . 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 7122 . . . . . . . 8 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
2524adantr 481 . . . . . . 7 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹𝑓 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
2625fneq1d 5949 . . . . . 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 247 . . . . 5 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹𝑓 𝑅𝐺) Fn (dom 𝐹 ∩ dom 𝐺))
28 fndm 5958 . . . . 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 3642 . . . 4 (dom (𝐹𝑓 𝑅𝐺) = (dom 𝐹 ∩ dom 𝐺) → dom (𝐹𝑓 𝑅𝐺) ⊆ (dom 𝐹 ∩ dom 𝐺))
31 cores2 5617 . . . 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 5937 . . . . 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 5254 . . 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 2657 . 2 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝑓 𝑅𝐺) ∘ (𝐻 ↾ (𝐻 “ (dom 𝐹 ∩ dom 𝐺)))))
37 simpr2 1066 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐹𝐻) ∈ V)
38 simpr3 1067 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (𝐺𝐻) ∈ V)
39 offval3 7122 . . . 4 (((𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V) → ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)) = (𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ↦ (((𝐹𝐻)‘𝑥)𝑅((𝐺𝐻)‘𝑥))))
4037, 38, 39syl2anc 692 . . 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 6308 . . . . . 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 5612 . . . . . 6 dom (𝐹𝐻) = (𝐻 “ dom 𝐹)
44 dmco 5612 . . . . . 6 dom (𝐺𝐻) = (𝐻 “ dom 𝐺)
4543, 44ineq12i 3796 . . . . 5 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) = ((𝐻 “ dom 𝐹) ∩ (𝐻 “ dom 𝐺))
4642, 45syl6reqr 2674 . . . 4 (((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) = (𝐻 “ (dom 𝐹 ∩ dom 𝐺)))
47 simplr1 1101 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → Fun 𝐻)
48 inss2 3818 . . . . . . . . 9 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom (𝐺𝐻)
49 dmcoss 5355 . . . . . . . . 9 dom (𝐺𝐻) ⊆ dom 𝐻
5048, 49sstri 3597 . . . . . . . 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 3588 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → 𝑥 ∈ dom 𝐻)
53 fvco 6241 . . . . . 6 ((Fun 𝐻𝑥 ∈ dom 𝐻) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
5447, 52, 53syl2anc 692 . . . . 5 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
55 inss1 3817 . . . . . . . . 9 (dom (𝐹𝐻) ∩ dom (𝐺𝐻)) ⊆ dom (𝐹𝐻)
56 dmcoss 5355 . . . . . . . . 9 dom (𝐹𝐻) ⊆ dom 𝐻
5755, 56sstri 3597 . . . . . . . 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 3588 . . . . . 6 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → 𝑥 ∈ dom 𝐻)
60 fvco 6241 . . . . . 6 ((Fun 𝐻𝑥 ∈ dom 𝐻) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
6147, 59, 60syl2anc 692 . . . . 5 ((((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) ∧ 𝑥 ∈ (dom (𝐹𝐻) ∩ dom (𝐺𝐻))) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
6254, 61oveq12d 6633 . . . 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 4702 . . 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 2655 . 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 2665 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 1036   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  cin 3559  wss 3560  cmpt 4683  ccnv 5083  dom cdm 5084  cres 5086  cima 5087  ccom 5088  Fun wfun 5851   Fn wfn 5852  cfv 5857  (class class class)co 6615  𝑓 cof 6860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862
This theorem is referenced by:  oftpos  20198
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