Theorem ofco 6103

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 5653	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐻‘𝑥) ∈ 𝐶)
3	1	feqmptd 5571	. . 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 2178	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
10		eqidd 2178	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐺‘𝑦) = (𝐺‘𝑦))
11	4, 5, 6, 7, 8, 9, 10	offval 6092	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑦 ∈ 𝐶 ↦ ((𝐹‘𝑦)𝑅(𝐺‘𝑦))))
12		fveq2 5517	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐻‘𝑥)))
13		fveq2 5517	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐺‘𝑦) = (𝐺‘(𝐻‘𝑥)))
14	12, 13	oveq12d 5895	. . 3 ⊢ (𝑦 = (𝐻‘𝑥) → ((𝐹‘𝑦)𝑅(𝐺‘𝑦)) = ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥))))
15	2, 3, 11, 14	fmptco 5684	. 2 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) ∘ 𝐻) = (𝑥 ∈ 𝐷 ↦ ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))))
16		inss1 3357	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
17	8, 16	eqsstrri 3190	. . . . 5 ⊢ 𝐶 ⊆ 𝐴
18		fss 5379	. . . . 5 ⊢ ((𝐻:𝐷⟶𝐶 ∧ 𝐶 ⊆ 𝐴) → 𝐻:𝐷⟶𝐴)
19	1, 17, 18	sylancl 413	. . . 4 ⊢ (𝜑 → 𝐻:𝐷⟶𝐴)
20		fnfco 5392	. . . 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 3358	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
23	8, 22	eqsstrri 3190	. . . . 5 ⊢ 𝐶 ⊆ 𝐵
24		fss 5379	. . . . 5 ⊢ ((𝐻:𝐷⟶𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐻:𝐷⟶𝐵)
25	1, 23, 24	sylancl 413	. . . 4 ⊢ (𝜑 → 𝐻:𝐷⟶𝐵)
26		fnfco 5392	. . . 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 3346	. . 3 ⊢ (𝐷 ∩ 𝐷) = 𝐷
30		ffn 5367	. . . . 5 ⊢ (𝐻:𝐷⟶𝐶 → 𝐻 Fn 𝐷)
31	1, 30	syl 14	. . . 4 ⊢ (𝜑 → 𝐻 Fn 𝐷)
32		fvco2 5587	. . . 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 5587	. . . 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 6092	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) ∘𝑓 𝑅(𝐺 ∘ 𝐻)) = (𝑥 ∈ 𝐷 ↦ ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))))
37	15, 36	eqtr4d 2213	1 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘𝑓 𝑅(𝐺 ∘ 𝐻)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148   ∩ cin 3130   ⊆ wss 3131   ↦ cmpt 4066   ∘ ccom 4632   Fn wfn 5213  ⟶wf 5214  ‘cfv 5218  (class class class)co 5877   ∘_𝑓 cof 6083

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-of 6085

This theorem is referenced by: (None)