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Theorem ofco 7085
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 6519 . . 3 ((𝜑𝑥𝐷) → (𝐻𝑥) ∈ 𝐶)
31feqmptd 6408 . . 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 2775 . . . 4 ((𝜑𝑦𝐴) → (𝐹𝑦) = (𝐹𝑦))
10 eqidd 2775 . . . 4 ((𝜑𝑦𝐵) → (𝐺𝑦) = (𝐺𝑦))
114, 5, 6, 7, 8, 9, 10offval 7072 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑦𝐶 ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
12 fveq2 6348 . . . 4 (𝑦 = (𝐻𝑥) → (𝐹𝑦) = (𝐹‘(𝐻𝑥)))
13 fveq2 6348 . . . 4 (𝑦 = (𝐻𝑥) → (𝐺𝑦) = (𝐺‘(𝐻𝑥)))
1412, 13oveq12d 6830 . . 3 (𝑦 = (𝐻𝑥) → ((𝐹𝑦)𝑅(𝐺𝑦)) = ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥))))
152, 3, 11, 14fmptco 6557 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = (𝑥𝐷 ↦ ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥)))))
16 inss1 3988 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
178, 16eqsstr3i 3792 . . . . 5 𝐶𝐴
18 fss 6209 . . . . 5 ((𝐻:𝐷𝐶𝐶𝐴) → 𝐻:𝐷𝐴)
191, 17, 18sylancl 575 . . . 4 (𝜑𝐻:𝐷𝐴)
20 fnfco 6224 . . . 4 ((𝐹 Fn 𝐴𝐻:𝐷𝐴) → (𝐹𝐻) Fn 𝐷)
214, 19, 20syl2anc 574 . . 3 (𝜑 → (𝐹𝐻) Fn 𝐷)
22 inss2 3989 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
238, 22eqsstr3i 3792 . . . . 5 𝐶𝐵
24 fss 6209 . . . . 5 ((𝐻:𝐷𝐶𝐶𝐵) → 𝐻:𝐷𝐵)
251, 23, 24sylancl 575 . . . 4 (𝜑𝐻:𝐷𝐵)
26 fnfco 6224 . . . 4 ((𝐺 Fn 𝐵𝐻:𝐷𝐵) → (𝐺𝐻) Fn 𝐷)
275, 25, 26syl2anc 574 . . 3 (𝜑 → (𝐺𝐻) Fn 𝐷)
28 ofco.6 . . 3 (𝜑𝐷𝑋)
29 inidm 3978 . . 3 (𝐷𝐷) = 𝐷
301ffnd 6197 . . . 4 (𝜑𝐻 Fn 𝐷)
31 fvco2 6432 . . . 4 ((𝐻 Fn 𝐷𝑥𝐷) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
3230, 31sylan 570 . . 3 ((𝜑𝑥𝐷) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
33 fvco2 6432 . . . 4 ((𝐻 Fn 𝐷𝑥𝐷) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
3430, 33sylan 570 . . 3 ((𝜑𝑥𝐷) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
3521, 27, 28, 28, 29, 32, 34offval 7072 . 2 (𝜑 → ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)) = (𝑥𝐷 ↦ ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥)))))
3615, 35eqtr4d 2811 1 (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1634  wcel 2148  cin 3728  wss 3729  cmpt 4876  ccom 5267   Fn wfn 6037  wf 6038  cfv 6042  (class class class)co 6812  𝑓 cof 7063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-of 7065
This theorem is referenced by:  gsumzaddlem  18548  coe1add  19869  pf1ind  19954  mendring  38303
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