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Theorem ofco 5873
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 5434 . . 3 ((𝜑𝑥𝐷) → (𝐻𝑥) ∈ 𝐶)
31feqmptd 5357 . . 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 2089 . . . 4 ((𝜑𝑦𝐴) → (𝐹𝑦) = (𝐹𝑦))
10 eqidd 2089 . . . 4 ((𝜑𝑦𝐵) → (𝐺𝑦) = (𝐺𝑦))
114, 5, 6, 7, 8, 9, 10offval 5863 . . 3 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑦𝐶 ↦ ((𝐹𝑦)𝑅(𝐺𝑦))))
12 fveq2 5305 . . . 4 (𝑦 = (𝐻𝑥) → (𝐹𝑦) = (𝐹‘(𝐻𝑥)))
13 fveq2 5305 . . . 4 (𝑦 = (𝐻𝑥) → (𝐺𝑦) = (𝐺‘(𝐻𝑥)))
1412, 13oveq12d 5670 . . 3 (𝑦 = (𝐻𝑥) → ((𝐹𝑦)𝑅(𝐺𝑦)) = ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥))))
152, 3, 11, 14fmptco 5464 . 2 (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = (𝑥𝐷 ↦ ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥)))))
16 inss1 3220 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
178, 16eqsstr3i 3057 . . . . 5 𝐶𝐴
18 fss 5172 . . . . 5 ((𝐻:𝐷𝐶𝐶𝐴) → 𝐻:𝐷𝐴)
191, 17, 18sylancl 404 . . . 4 (𝜑𝐻:𝐷𝐴)
20 fnfco 5185 . . . 4 ((𝐹 Fn 𝐴𝐻:𝐷𝐴) → (𝐹𝐻) Fn 𝐷)
214, 19, 20syl2anc 403 . . 3 (𝜑 → (𝐹𝐻) Fn 𝐷)
22 inss2 3221 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
238, 22eqsstr3i 3057 . . . . 5 𝐶𝐵
24 fss 5172 . . . . 5 ((𝐻:𝐷𝐶𝐶𝐵) → 𝐻:𝐷𝐵)
251, 23, 24sylancl 404 . . . 4 (𝜑𝐻:𝐷𝐵)
26 fnfco 5185 . . . 4 ((𝐺 Fn 𝐵𝐻:𝐷𝐵) → (𝐺𝐻) Fn 𝐷)
275, 25, 26syl2anc 403 . . 3 (𝜑 → (𝐺𝐻) Fn 𝐷)
28 ofco.6 . . 3 (𝜑𝐷𝑋)
29 inidm 3209 . . 3 (𝐷𝐷) = 𝐷
30 ffn 5161 . . . . 5 (𝐻:𝐷𝐶𝐻 Fn 𝐷)
311, 30syl 14 . . . 4 (𝜑𝐻 Fn 𝐷)
32 fvco2 5373 . . . 4 ((𝐻 Fn 𝐷𝑥𝐷) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
3331, 32sylan 277 . . 3 ((𝜑𝑥𝐷) → ((𝐹𝐻)‘𝑥) = (𝐹‘(𝐻𝑥)))
34 fvco2 5373 . . . 4 ((𝐻 Fn 𝐷𝑥𝐷) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
3531, 34sylan 277 . . 3 ((𝜑𝑥𝐷) → ((𝐺𝐻)‘𝑥) = (𝐺‘(𝐻𝑥)))
3621, 27, 28, 28, 29, 33, 35offval 5863 . 2 (𝜑 → ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)) = (𝑥𝐷 ↦ ((𝐹‘(𝐻𝑥))𝑅(𝐺‘(𝐻𝑥)))))
3715, 36eqtr4d 2123 1 (𝜑 → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  cin 2998  wss 2999  cmpt 3899  ccom 4442   Fn wfn 5010  wf 5011  cfv 5015  (class class class)co 5652  𝑓 cof 5854
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-of 5856
This theorem is referenced by: (None)
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