Theorem ofoprabco 33082

Description: Function operation as a composition with an operation. (Contributed by Thierry Arnoux, 4-Jun-2017.)

Hypotheses

Ref	Expression
ofoprabco.1	⊢ Ⅎ𝑎𝑀
ofoprabco.2	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
ofoprabco.3	⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
ofoprabco.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofoprabco.5	⊢ (𝜑 → 𝑀 = (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩))
ofoprabco.6	⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)))

Assertion

Ref	Expression
ofoprabco	⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑁 ∘ 𝑀))

Distinct variable groups:   𝑥,𝑎,𝑦,𝐴   𝐵,𝑎,𝑥,𝑦   𝐶,𝑎,𝑥,𝑦   𝐹,𝑎,𝑥,𝑦   𝐺,𝑎,𝑥,𝑦   𝑁,𝑎   𝑅,𝑎,𝑥,𝑦   𝜑,𝑎,𝑥,𝑦

Allowed substitution hints:   𝑀(𝑥, 𝑦, 𝑎)   𝑁(𝑥, 𝑦)   𝑉(𝑥, 𝑦, 𝑎)

Proof of Theorem ofoprabco

Step	Hyp	Ref	Expression
1		ofoprabco.5	. . . . . 6 ⊢ (𝜑 → 𝑀 = (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩))
2		ofoprabco.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
3	2	ffvelcdmda 7083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ 𝐵)
4		ofoprabco.3	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
5	4	ffvelcdmda 7083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ 𝐶)
6		opelxpi 5700	. . . . . . 7 ⊢ (((𝐹‘𝑎) ∈ 𝐵 ∧ (𝐺‘𝑎) ∈ 𝐶) → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ (𝐵 × 𝐶))
7	3, 5, 6	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ (𝐵 × 𝐶))
8	1, 7	fvmpt2d 7007	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑀‘𝑎) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
9	8	fveq2d 6889	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑁‘(𝑀‘𝑎)) = (𝑁‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩))
10		df-ov 7422	. . . . 5 ⊢ ((𝐹‘𝑎)𝑁(𝐺‘𝑎)) = (𝑁‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
11	10	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎)𝑁(𝐺‘𝑎)) = (𝑁‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩))
12		ofoprabco.6	. . . . . 6 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)))
13	12	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)))
14		simprl 783	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐺‘𝑎))) → 𝑥 = (𝐹‘𝑎))
15		simprr 785	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐺‘𝑎))) → 𝑦 = (𝐺‘𝑎))
16	14, 15	oveq12d 7437	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐺‘𝑎))) → (𝑥𝑅𝑦) = ((𝐹‘𝑎)𝑅(𝐺‘𝑎)))
17		ovexd 7454	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎)𝑅(𝐺‘𝑎)) ∈ V)
18	13, 16, 3, 5, 17	ovmpod 7571	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎)𝑁(𝐺‘𝑎)) = ((𝐹‘𝑎)𝑅(𝐺‘𝑎)))
19	9, 11, 18	3eqtr2d 2806	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑁‘(𝑀‘𝑎)) = ((𝐹‘𝑎)𝑅(𝐺‘𝑎)))
20	19	mpteq2dva 5206	. 2 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ (𝑁‘(𝑀‘𝑎))) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎)𝑅(𝐺‘𝑎))))
21		ovex 7452	. . . . . 6 ⊢ (𝑥𝑅𝑦) ∈ V
22	21	rgen2w 3086	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦) ∈ V
23		eqid 2765	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦))
24	23	fmpo 8071	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦) ∈ V ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V)
25	22, 24	mpbi 233	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V
26	12	feq1d 6691	. . . 4 ⊢ (𝜑 → (𝑁:(𝐵 × 𝐶)⟶V ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V))
27	25, 26	mpbiri 261	. . 3 ⊢ (𝜑 → 𝑁:(𝐵 × 𝐶)⟶V)
28	1, 7	fmpt3d 7115	. . 3 ⊢ (𝜑 → 𝑀:𝐴⟶(𝐵 × 𝐶))
29		ofoprabco.1	. . . 4 ⊢ Ⅎ𝑎𝑀
30	29	fcomptf 33076	. . 3 ⊢ ((𝑁:(𝐵 × 𝐶)⟶V ∧ 𝑀:𝐴⟶(𝐵 × 𝐶)) → (𝑁 ∘ 𝑀) = (𝑎 ∈ 𝐴 ↦ (𝑁‘(𝑀‘𝑎))))
31	27, 28, 30	syl2anc 596	. 2 ⊢ (𝜑 → (𝑁 ∘ 𝑀) = (𝑎 ∈ 𝐴 ↦ (𝑁‘(𝑀‘𝑎))))
32		ofoprabco.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
33	2	feqmptd 6953	. . 3 ⊢ (𝜑 → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
34	4	feqmptd 6953	. . 3 ⊢ (𝜑 → 𝐺 = (𝑎 ∈ 𝐴 ↦ (𝐺‘𝑎)))
35	32, 3, 5, 33, 34	offval2 7704	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎)𝑅(𝐺‘𝑎))))
36	20, 31, 35	3eqtr4rd 2811	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑁 ∘ 𝑀))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Ⅎwnfc 2912  ∀wral 3081  Vcvv 3457  ⟨cop 4597   ↦ cmpt 5194   × cxp 5661   ∘ ccom 5667  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421   ∘_f cof 7682

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7684  df-1st 7992  df-2nd 7993

This theorem is used by:  ofpreima  33083  rrvadd  34909