Theorem ofco 6151

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	ffvelcdmda 5694	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐻‘𝑥) ∈ 𝐶)
3	1	feqmptd 5611	. . 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 2194	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
10		eqidd 2194	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐺‘𝑦) = (𝐺‘𝑦))
11	4, 5, 6, 7, 8, 9, 10	offval 6140	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑦 ∈ 𝐶 ↦ ((𝐹‘𝑦)𝑅(𝐺‘𝑦))))
12		fveq2 5555	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐻‘𝑥)))
13		fveq2 5555	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐺‘𝑦) = (𝐺‘(𝐻‘𝑥)))
14	12, 13	oveq12d 5937	. . 3 ⊢ (𝑦 = (𝐻‘𝑥) → ((𝐹‘𝑦)𝑅(𝐺‘𝑦)) = ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥))))
15	2, 3, 11, 14	fmptco 5725	. 2 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) ∘ 𝐻) = (𝑥 ∈ 𝐷 ↦ ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))))
16		inss1 3380	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
17	8, 16	eqsstrri 3213	. . . . 5 ⊢ 𝐶 ⊆ 𝐴
18		fss 5416	. . . . 5 ⊢ ((𝐻:𝐷⟶𝐶 ∧ 𝐶 ⊆ 𝐴) → 𝐻:𝐷⟶𝐴)
19	1, 17, 18	sylancl 413	. . . 4 ⊢ (𝜑 → 𝐻:𝐷⟶𝐴)
20		fnfco 5429	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐻:𝐷⟶𝐴) → (𝐹 ∘ 𝐻) Fn 𝐷)
21	4, 19, 20	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐻) Fn 𝐷)
22		inss2 3381	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
23	8, 22	eqsstrri 3213	. . . . 5 ⊢ 𝐶 ⊆ 𝐵
24		fss 5416	. . . . 5 ⊢ ((𝐻:𝐷⟶𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐻:𝐷⟶𝐵)
25	1, 23, 24	sylancl 413	. . . 4 ⊢ (𝜑 → 𝐻:𝐷⟶𝐵)
26		fnfco 5429	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐻:𝐷⟶𝐵) → (𝐺 ∘ 𝐻) Fn 𝐷)
27	5, 25, 26	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐻) Fn 𝐷)
28		ofco.6	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
29		inidm 3369	. . 3 ⊢ (𝐷 ∩ 𝐷) = 𝐷
30		ffn 5404	. . . . 5 ⊢ (𝐻:𝐷⟶𝐶 → 𝐻 Fn 𝐷)
31	1, 30	syl 14	. . . 4 ⊢ (𝜑 → 𝐻 Fn 𝐷)
32		fvco2 5627	. . . 4 ⊢ ((𝐻 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
33	31, 32	sylan 283	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
34		fvco2 5627	. . . 4 ⊢ ((𝐻 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
35	31, 34	sylan 283	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
36	21, 27, 28, 28, 29, 33, 35	offval 6140	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) ∘𝑓 𝑅(𝐺 ∘ 𝐻)) = (𝑥 ∈ 𝐷 ↦ ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))))
37	15, 36	eqtr4d 2229	1 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘𝑓 𝑅(𝐺 ∘ 𝐻)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164   ∩ cin 3153   ⊆ wss 3154   ↦ cmpt 4091   ∘ ccom 4664   Fn wfn 5250  ⟶wf 5251  ‘cfv 5255  (class class class)co 5919   ∘_𝑓 cof 6130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-of 6132

This theorem is referenced by: (None)