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	⊢ (φ → 𝐹 Fn A)
ofco.2	⊢ (φ → 𝐺 Fn B)
ofco.3	⊢ (φ → 𝐻:𝐷⟶𝐶)
ofco.4	⊢ (φ → A ∈ 𝑉)
ofco.5	⊢ (φ → B ∈ 𝑊)
ofco.6	⊢ (φ → 𝐷 ∈ 𝑋)
ofco.7	⊢ (A ∩ B) = 𝐶

Assertion

Ref	Expression
ofco	⊢ (φ → ((𝐹 ∘𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘𝑓 𝑅(𝐺 ∘ 𝐻)))

Proof of Theorem ofco

Dummy variables y x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofco.3	. . . 4 ⊢ (φ → 𝐻:𝐷⟶𝐶)
2	1	ffvelrnda 5245	. . 3 ⊢ ((φ ∧ x ∈ 𝐷) → (𝐻‘x) ∈ 𝐶)
3	1	feqmptd 5169	. . 3 ⊢ (φ → 𝐻 = (x ∈ 𝐷 ↦ (𝐻‘x)))
4		ofco.1	. . . 4 ⊢ (φ → 𝐹 Fn A)
5		ofco.2	. . . 4 ⊢ (φ → 𝐺 Fn B)
6		ofco.4	. . . 4 ⊢ (φ → A ∈ 𝑉)
7		ofco.5	. . . 4 ⊢ (φ → B ∈ 𝑊)
8		ofco.7	. . . 4 ⊢ (A ∩ B) = 𝐶
9		eqidd 2038	. . . 4 ⊢ ((φ ∧ y ∈ A) → (𝐹‘y) = (𝐹‘y))
10		eqidd 2038	. . . 4 ⊢ ((φ ∧ y ∈ B) → (𝐺‘y) = (𝐺‘y))
11	4, 5, 6, 7, 8, 9, 10	offval 5661	. . 3 ⊢ (φ → (𝐹 ∘𝑓 𝑅𝐺) = (y ∈ 𝐶 ↦ ((𝐹‘y)𝑅(𝐺‘y))))
12		fveq2 5121	. . . 4 ⊢ (y = (𝐻‘x) → (𝐹‘y) = (𝐹‘(𝐻‘x)))
13		fveq2 5121	. . . 4 ⊢ (y = (𝐻‘x) → (𝐺‘y) = (𝐺‘(𝐻‘x)))
14	12, 13	oveq12d 5473	. . 3 ⊢ (y = (𝐻‘x) → ((𝐹‘y)𝑅(𝐺‘y)) = ((𝐹‘(𝐻‘x))𝑅(𝐺‘(𝐻‘x))))
15	2, 3, 11, 14	fmptco 5273	. 2 ⊢ (φ → ((𝐹 ∘𝑓 𝑅𝐺) ∘ 𝐻) = (x ∈ 𝐷 ↦ ((𝐹‘(𝐻‘x))𝑅(𝐺‘(𝐻‘x)))))
16		inss1 3151	. . . . . 6 ⊢ (A ∩ B) ⊆ A
17	8, 16	eqsstr3i 2970	. . . . 5 ⊢ 𝐶 ⊆ A
18		fss 4997	. . . . 5 ⊢ ((𝐻:𝐷⟶𝐶 ∧ 𝐶 ⊆ A) → 𝐻:𝐷⟶A)
19	1, 17, 18	sylancl 392	. . . 4 ⊢ (φ → 𝐻:𝐷⟶A)
20		fnfco 5008	. . . 4 ⊢ ((𝐹 Fn A ∧ 𝐻:𝐷⟶A) → (𝐹 ∘ 𝐻) Fn 𝐷)
21	4, 19, 20	syl2anc 391	. . 3 ⊢ (φ → (𝐹 ∘ 𝐻) Fn 𝐷)
22		inss2 3152	. . . . . 6 ⊢ (A ∩ B) ⊆ B
23	8, 22	eqsstr3i 2970	. . . . 5 ⊢ 𝐶 ⊆ B
24		fss 4997	. . . . 5 ⊢ ((𝐻:𝐷⟶𝐶 ∧ 𝐶 ⊆ B) → 𝐻:𝐷⟶B)
25	1, 23, 24	sylancl 392	. . . 4 ⊢ (φ → 𝐻:𝐷⟶B)
26		fnfco 5008	. . . 4 ⊢ ((𝐺 Fn B ∧ 𝐻:𝐷⟶B) → (𝐺 ∘ 𝐻) Fn 𝐷)
27	5, 25, 26	syl2anc 391	. . 3 ⊢ (φ → (𝐺 ∘ 𝐻) Fn 𝐷)
28		ofco.6	. . 3 ⊢ (φ → 𝐷 ∈ 𝑋)
29		inidm 3140	. . 3 ⊢ (𝐷 ∩ 𝐷) = 𝐷
30		ffn 4989	. . . . 5 ⊢ (𝐻:𝐷⟶𝐶 → 𝐻 Fn 𝐷)
31	1, 30	syl 14	. . . 4 ⊢ (φ → 𝐻 Fn 𝐷)
32		fvco2 5185	. . . 4 ⊢ ((𝐻 Fn 𝐷 ∧ x ∈ 𝐷) → ((𝐹 ∘ 𝐻)‘x) = (𝐹‘(𝐻‘x)))
33	31, 32	sylan 267	. . 3 ⊢ ((φ ∧ x ∈ 𝐷) → ((𝐹 ∘ 𝐻)‘x) = (𝐹‘(𝐻‘x)))
34		fvco2 5185	. . . 4 ⊢ ((𝐻 Fn 𝐷 ∧ x ∈ 𝐷) → ((𝐺 ∘ 𝐻)‘x) = (𝐺‘(𝐻‘x)))
35	31, 34	sylan 267	. . 3 ⊢ ((φ ∧ x ∈ 𝐷) → ((𝐺 ∘ 𝐻)‘x) = (𝐺‘(𝐻‘x)))
36	21, 27, 28, 28, 29, 33, 35	offval 5661	. 2 ⊢ (φ → ((𝐹 ∘ 𝐻) ∘𝑓 𝑅(𝐺 ∘ 𝐻)) = (x ∈ 𝐷 ↦ ((𝐹‘(𝐻‘x))𝑅(𝐺‘(𝐻‘x)))))
37	15, 36	eqtr4d 2072	1 ⊢ (φ → ((𝐹 ∘𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘𝑓 𝑅(𝐺 ∘ 𝐻)))

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)