Theorem ofoprabco 32682

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 7118	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ 𝐵)
4		ofoprabco.3	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
5	4	ffvelcdmda 7118	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ 𝐶)
6		opelxpi 5737	. . . . . . 7 ⊢ (((𝐹‘𝑎) ∈ 𝐵 ∧ (𝐺‘𝑎) ∈ 𝐶) → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ (𝐵 × 𝐶))
7	3, 5, 6	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ (𝐵 × 𝐶))
8	1, 7	fvmpt2d 7042	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑀‘𝑎) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
9	8	fveq2d 6924	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑁‘(𝑀‘𝑎)) = (𝑁‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩))
10		df-ov 7451	. . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)))
14		simprl 770	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐺‘𝑎))) → 𝑥 = (𝐹‘𝑎))
15		simprr 772	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐺‘𝑎))) → 𝑦 = (𝐺‘𝑎))
16	14, 15	oveq12d 7466	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐺‘𝑎))) → (𝑥𝑅𝑦) = ((𝐹‘𝑎)𝑅(𝐺‘𝑎)))
17		ovexd 7483	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎)𝑅(𝐺‘𝑎)) ∈ V)
18	13, 16, 3, 5, 17	ovmpod 7602	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎)𝑁(𝐺‘𝑎)) = ((𝐹‘𝑎)𝑅(𝐺‘𝑎)))
19	9, 11, 18	3eqtr2d 2786	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑁‘(𝑀‘𝑎)) = ((𝐹‘𝑎)𝑅(𝐺‘𝑎)))
20	19	mpteq2dva 5266	. 2 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ (𝑁‘(𝑀‘𝑎))) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎)𝑅(𝐺‘𝑎))))
21		ovex 7481	. . . . . 6 ⊢ (𝑥𝑅𝑦) ∈ V
22	21	rgen2w 3072	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦) ∈ V
23		eqid 2740	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦))
24	23	fmpo 8109	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦) ∈ V ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V)
25	22, 24	mpbi 230	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V
26	12	feq1d 6732	. . . 4 ⊢ (𝜑 → (𝑁:(𝐵 × 𝐶)⟶V ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V))
27	25, 26	mpbiri 258	. . 3 ⊢ (𝜑 → 𝑁:(𝐵 × 𝐶)⟶V)
28	1, 7	fmpt3d 7150	. . 3 ⊢ (𝜑 → 𝑀:𝐴⟶(𝐵 × 𝐶))
29		ofoprabco.1	. . . 4 ⊢ Ⅎ𝑎𝑀
30	29	fcomptf 32676	. . 3 ⊢ ((𝑁:(𝐵 × 𝐶)⟶V ∧ 𝑀:𝐴⟶(𝐵 × 𝐶)) → (𝑁 ∘ 𝑀) = (𝑎 ∈ 𝐴 ↦ (𝑁‘(𝑀‘𝑎))))
31	27, 28, 30	syl2anc 583	. 2 ⊢ (𝜑 → (𝑁 ∘ 𝑀) = (𝑎 ∈ 𝐴 ↦ (𝑁‘(𝑀‘𝑎))))
32		ofoprabco.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
33	2	feqmptd 6990	. . 3 ⊢ (𝜑 → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
34	4	feqmptd 6990	. . 3 ⊢ (𝜑 → 𝐺 = (𝑎 ∈ 𝐴 ↦ (𝐺‘𝑎)))
35	32, 3, 5, 33, 34	offval2 7734	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎)𝑅(𝐺‘𝑎))))
36	20, 31, 35	3eqtr4rd 2791	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑁 ∘ 𝑀))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Ⅎwnfc 2893  ∀wral 3067  Vcvv 3488  ⟨cop 4654   ↦ cmpt 5249   × cxp 5698   ∘ ccom 5704  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450   ∘_f cof 7712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-1st 8030  df-2nd 8031

This theorem is referenced by:  ofpreima  32683  rrvadd  34417