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Theorem ofco 5671
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 (φA 𝑉)
ofco.5 (φB 𝑊)
ofco.6 (φ𝐷 𝑋)
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 5245 . . 3 ((φ x 𝐷) → (𝐻x) 𝐶)
31feqmptd 5169 . . 3 (φ𝐻 = (x 𝐷 ↦ (𝐻x)))
4 ofco.1 . . . 4 (φ𝐹 Fn A)
5 ofco.2 . . . 4 (φ𝐺 Fn B)
6 ofco.4 . . . 4 (φA 𝑉)
7 ofco.5 . . . 4 (φB 𝑊)
8 ofco.7 . . . 4 (AB) = 𝐶
9 eqidd 2038 . . . 4 ((φ y A) → (𝐹y) = (𝐹y))
10 eqidd 2038 . . . 4 ((φ y B) → (𝐺y) = (𝐺y))
114, 5, 6, 7, 8, 9, 10offval 5661 . . 3 (φ → (𝐹𝑓 𝑅𝐺) = (y 𝐶 ↦ ((𝐹y)𝑅(𝐺y))))
12 fveq2 5121 . . . 4 (y = (𝐻x) → (𝐹y) = (𝐹‘(𝐻x)))
13 fveq2 5121 . . . 4 (y = (𝐻x) → (𝐺y) = (𝐺‘(𝐻x)))
1412, 13oveq12d 5473 . . 3 (y = (𝐻x) → ((𝐹y)𝑅(𝐺y)) = ((𝐹‘(𝐻x))𝑅(𝐺‘(𝐻x))))
152, 3, 11, 14fmptco 5273 . 2 (φ → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = (x 𝐷 ↦ ((𝐹‘(𝐻x))𝑅(𝐺‘(𝐻x)))))
16 inss1 3151 . . . . . 6 (AB) ⊆ A
178, 16eqsstr3i 2970 . . . . 5 𝐶A
18 fss 4997 . . . . 5 ((𝐻:𝐷𝐶 𝐶A) → 𝐻:𝐷A)
191, 17, 18sylancl 392 . . . 4 (φ𝐻:𝐷A)
20 fnfco 5008 . . . 4 ((𝐹 Fn A 𝐻:𝐷A) → (𝐹𝐻) Fn 𝐷)
214, 19, 20syl2anc 391 . . 3 (φ → (𝐹𝐻) Fn 𝐷)
22 inss2 3152 . . . . . 6 (AB) ⊆ B
238, 22eqsstr3i 2970 . . . . 5 𝐶B
24 fss 4997 . . . . 5 ((𝐻:𝐷𝐶 𝐶B) → 𝐻:𝐷B)
251, 23, 24sylancl 392 . . . 4 (φ𝐻:𝐷B)
26 fnfco 5008 . . . 4 ((𝐺 Fn B 𝐻:𝐷B) → (𝐺𝐻) Fn 𝐷)
275, 25, 26syl2anc 391 . . 3 (φ → (𝐺𝐻) Fn 𝐷)
28 ofco.6 . . 3 (φ𝐷 𝑋)
29 inidm 3140 . . 3 (𝐷𝐷) = 𝐷
30 ffn 4989 . . . . 5 (𝐻:𝐷𝐶𝐻 Fn 𝐷)
311, 30syl 14 . . . 4 (φ𝐻 Fn 𝐷)
32 fvco2 5185 . . . 4 ((𝐻 Fn 𝐷 x 𝐷) → ((𝐹𝐻)‘x) = (𝐹‘(𝐻x)))
3331, 32sylan 267 . . 3 ((φ x 𝐷) → ((𝐹𝐻)‘x) = (𝐹‘(𝐻x)))
34 fvco2 5185 . . . 4 ((𝐻 Fn 𝐷 x 𝐷) → ((𝐺𝐻)‘x) = (𝐺‘(𝐻x)))
3531, 34sylan 267 . . 3 ((φ x 𝐷) → ((𝐺𝐻)‘x) = (𝐺‘(𝐻x)))
3621, 27, 28, 28, 29, 33, 35offval 5661 . 2 (φ → ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)) = (x 𝐷 ↦ ((𝐹‘(𝐻x))𝑅(𝐺‘(𝐻x)))))
3715, 36eqtr4d 2072 1 (φ → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  cin 2910  wss 2911  cmpt 3809  ccom 4292   Fn wfn 4840  wf 4841  cfv 4845  (class class class)co 5455  𝑓 cof 5652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-of 5654
This theorem is referenced by: (None)
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