Theorem ofoprabco 30409

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	ffvelrnda 6851	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ 𝐵)
4		ofoprabco.3	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
5	4	ffvelrnda 6851	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ 𝐶)
6		opelxpi 5592	. . . . . . 7 ⊢ (((𝐹‘𝑎) ∈ 𝐵 ∧ (𝐺‘𝑎) ∈ 𝐶) → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ (𝐵 × 𝐶))
7	3, 5, 6	syl2anc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ (𝐵 × 𝐶))
8	1, 7	fvmpt2d 6781	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑀‘𝑎) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
9	8	fveq2d 6674	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑁‘(𝑀‘𝑎)) = (𝑁‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩))
10		df-ov 7159	. . . . 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 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)))
14		simprl 769	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐺‘𝑎))) → 𝑥 = (𝐹‘𝑎))
15		simprr 771	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐺‘𝑎))) → 𝑦 = (𝐺‘𝑎))
16	14, 15	oveq12d 7174	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐺‘𝑎))) → (𝑥𝑅𝑦) = ((𝐹‘𝑎)𝑅(𝐺‘𝑎)))
17		ovexd 7191	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎)𝑅(𝐺‘𝑎)) ∈ V)
18	13, 16, 3, 5, 17	ovmpod 7302	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎)𝑁(𝐺‘𝑎)) = ((𝐹‘𝑎)𝑅(𝐺‘𝑎)))
19	9, 11, 18	3eqtr2d 2862	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑁‘(𝑀‘𝑎)) = ((𝐹‘𝑎)𝑅(𝐺‘𝑎)))
20	19	mpteq2dva 5161	. 2 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ (𝑁‘(𝑀‘𝑎))) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎)𝑅(𝐺‘𝑎))))
21		ovex 7189	. . . . . 6 ⊢ (𝑥𝑅𝑦) ∈ V
22	21	rgen2w 3151	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦) ∈ V
23		eqid 2821	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦))
24	23	fmpo 7766	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦) ∈ V ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V)
25	22, 24	mpbi 232	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V
26	12	feq1d 6499	. . . 4 ⊢ (𝜑 → (𝑁:(𝐵 × 𝐶)⟶V ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V))
27	25, 26	mpbiri 260	. . 3 ⊢ (𝜑 → 𝑁:(𝐵 × 𝐶)⟶V)
28	1, 7	fmpt3d 6880	. . 3 ⊢ (𝜑 → 𝑀:𝐴⟶(𝐵 × 𝐶))
29		ofoprabco.1	. . . 4 ⊢ Ⅎ𝑎𝑀
30	29	fcomptf 30403	. . 3 ⊢ ((𝑁:(𝐵 × 𝐶)⟶V ∧ 𝑀:𝐴⟶(𝐵 × 𝐶)) → (𝑁 ∘ 𝑀) = (𝑎 ∈ 𝐴 ↦ (𝑁‘(𝑀‘𝑎))))
31	27, 28, 30	syl2anc 586	. 2 ⊢ (𝜑 → (𝑁 ∘ 𝑀) = (𝑎 ∈ 𝐴 ↦ (𝑁‘(𝑀‘𝑎))))
32		ofoprabco.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
33	2	feqmptd 6733	. . 3 ⊢ (𝜑 → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
34	4	feqmptd 6733	. . 3 ⊢ (𝜑 → 𝐺 = (𝑎 ∈ 𝐴 ↦ (𝐺‘𝑎)))
35	32, 3, 5, 33, 34	offval2 7426	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎)𝑅(𝐺‘𝑎))))
36	20, 31, 35	3eqtr4rd 2867	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑁 ∘ 𝑀))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Ⅎwnfc 2961  ∀wral 3138  Vcvv 3494  ⟨cop 4573   ↦ cmpt 5146   × cxp 5553   ∘ ccom 5559  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158   ∘_f cof 7407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-1st 7689  df-2nd 7690

This theorem is referenced by:  ofpreima  30410  rrvadd  31710