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Theorem ofco 6959
Description: The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
Hypotheses
Ref Expression
ofco.1 (𝜑𝐹 Fn 𝐴)
ofco.2 (𝜑𝐺 Fn 𝐵)
ofco.3 (𝜑𝐻:𝐷𝐶)
ofco.4 (𝜑𝐴𝑉)
ofco.5 (𝜑𝐵𝑊)
ofco.6 (𝜑𝐷𝑋)
ofco.7 (𝐴𝐵) = 𝐶
Assertion
Ref Expression
ofco (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)))

Proof of Theorem ofco
Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ofco.3 . . . 4 (𝜑𝐻:𝐷𝐶)
21ffvelrnda 6399 . . 3 ((𝜑𝑥𝐷) → (𝐻𝑥) ∈ 𝐶)
31feqmptd 6288 . . 3 (𝜑𝐻 = (𝑥𝐷 ↦ (𝐻𝑥)))
4 ofco.1 . . . 4 (𝜑𝐹 Fn 𝐴)
5 ofco.2 . . . 4 (𝜑𝐺 Fn 𝐵)
6 ofco.4 . . . 4 (𝜑𝐴𝑉)
7 ofco.5 . . . 4 (𝜑𝐵𝑊)
8 ofco.7 . . . 4 (𝐴𝐵) = 𝐶
9 eqidd 2652 . . . 4 ((𝜑𝑦𝐴) → (𝐹𝑦) = (𝐹𝑦))
10 eqidd 2652 . . . 4 ((𝜑𝑦𝐵) → (𝐺𝑦) = (𝐺𝑦))
114, 5, 6, 7, 8, 9, 10offval 6946 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑦𝐶 ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
12 fveq2 6229 . . . 4 (𝑦 = (𝐻𝑥) → (𝐹𝑦) = (𝐹‘(𝐻𝑥)))
13 fveq2 6229 . . . 4 (𝑦 = (𝐻𝑥) → (𝐺𝑦) = (𝐺‘(𝐻𝑥)))
1412, 13oveq12d 6708 . . 3 (𝑦 = (𝐻𝑥) → ((𝐹𝑦)𝑅(𝐺𝑦)) = ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥))))
152, 3, 11, 14fmptco 6436 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = (𝑥𝐷 ↦ ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥)))))
16 inss1 3866 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
178, 16eqsstr3i 3669 . . . . 5 𝐶𝐴
18 fss 6094 . . . . 5 ((𝐻:𝐷𝐶𝐶𝐴) → 𝐻:𝐷𝐴)
191, 17, 18sylancl 695 . . . 4 (𝜑𝐻:𝐷𝐴)
20 fnfco 6107 . . . 4 ((𝐹 Fn 𝐴𝐻:𝐷𝐴) → (𝐹𝐻) Fn 𝐷)
214, 19, 20syl2anc 694 . . 3 (𝜑 → (𝐹𝐻) Fn 𝐷)
22 inss2 3867 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
238, 22eqsstr3i 3669 . . . . 5 𝐶𝐵
24 fss 6094 . . . . 5 ((𝐻:𝐷𝐶𝐶𝐵) → 𝐻:𝐷𝐵)
251, 23, 24sylancl 695 . . . 4 (𝜑𝐻:𝐷𝐵)
26 fnfco 6107 . . . 4 ((𝐺 Fn 𝐵𝐻:𝐷𝐵) → (𝐺𝐻) Fn 𝐷)
275, 25, 26syl2anc 694 . . 3 (𝜑 → (𝐺𝐻) Fn 𝐷)
28 ofco.6 . . 3 (𝜑𝐷𝑋)
29 inidm 3855 . . 3 (𝐷𝐷) = 𝐷
30 ffn 6083 . . . . 5 (𝐻:𝐷𝐶𝐻 Fn 𝐷)
311, 30syl 17 . . . 4 (𝜑𝐻 Fn 𝐷)
32 fvco2 6312 . . . 4 ((𝐻 Fn 𝐷𝑥𝐷) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
3331, 32sylan 487 . . 3 ((𝜑𝑥𝐷) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
34 fvco2 6312 . . . 4 ((𝐻 Fn 𝐷𝑥𝐷) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
3531, 34sylan 487 . . 3 ((𝜑𝑥𝐷) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
3621, 27, 28, 28, 29, 33, 35offval 6946 . 2 (𝜑 → ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)) = (𝑥𝐷 ↦ ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥)))))
3715, 36eqtr4d 2688 1 (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cin 3606  wss 3607  cmpt 4762  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939
This theorem is referenced by:  gsumzaddlem  18367  coe1add  19682  pf1ind  19767  mendring  38079
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