Theorem ofoprabco 31286

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 7022	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ 𝐵)
4		ofoprabco.3	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)
5	4	ffvelcdmda 7022	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐺‘𝑎) ∈ 𝐶)
6		opelxpi 5662	. . . . . . 7 ⊢ (((𝐹‘𝑎) ∈ 𝐵 ∧ (𝐺‘𝑎) ∈ 𝐶) → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ (𝐵 × 𝐶))
7	3, 5, 6	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ (𝐵 × 𝐶))
8	1, 7	fvmpt2d 6949	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑀‘𝑎) = ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)
9	8	fveq2d 6834	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑁‘(𝑀‘𝑎)) = (𝑁‘⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩))
10		df-ov 7345	. . . . 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 482	. . . . 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 7360	. . . . 5 ⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ (𝑥 = (𝐹‘𝑎) ∧ 𝑦 = (𝐺‘𝑎))) → (𝑥𝑅𝑦) = ((𝐹‘𝑎)𝑅(𝐺‘𝑎)))
17		ovexd 7377	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎)𝑅(𝐺‘𝑎)) ∈ V)
18	13, 16, 3, 5, 17	ovmpod 7492	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝐹‘𝑎)𝑁(𝐺‘𝑎)) = ((𝐹‘𝑎)𝑅(𝐺‘𝑎)))
19	9, 11, 18	3eqtr2d 2783	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑁‘(𝑀‘𝑎)) = ((𝐹‘𝑎)𝑅(𝐺‘𝑎)))
20	19	mpteq2dva 5197	. 2 ⊢ (𝜑 → (𝑎 ∈ 𝐴 ↦ (𝑁‘(𝑀‘𝑎))) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎)𝑅(𝐺‘𝑎))))
21		ovex 7375	. . . . . 6 ⊢ (𝑥𝑅𝑦) ∈ V
22	21	rgen2w 3067	. . . . 5 ⊢ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦) ∈ V
23		eqid 2737	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦))
24	23	fmpo 7981	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦) ∈ V ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V)
25	22, 24	mpbi 229	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)):(𝐵 × 𝐶)⟶V
26	12	feq1d 6641	. . . 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 7051	. . 3 ⊢ (𝜑 → 𝑀:𝐴⟶(𝐵 × 𝐶))
29		ofoprabco.1	. . . 4 ⊢ Ⅎ𝑎𝑀
30	29	fcomptf 31280	. . 3 ⊢ ((𝑁:(𝐵 × 𝐶)⟶V ∧ 𝑀:𝐴⟶(𝐵 × 𝐶)) → (𝑁 ∘ 𝑀) = (𝑎 ∈ 𝐴 ↦ (𝑁‘(𝑀‘𝑎))))
31	27, 28, 30	syl2anc 585	. 2 ⊢ (𝜑 → (𝑁 ∘ 𝑀) = (𝑎 ∈ 𝐴 ↦ (𝑁‘(𝑀‘𝑎))))
32		ofoprabco.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
33	2	feqmptd 6898	. . 3 ⊢ (𝜑 → 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝐹‘𝑎)))
34	4	feqmptd 6898	. . 3 ⊢ (𝜑 → 𝐺 = (𝑎 ∈ 𝐴 ↦ (𝐺‘𝑎)))
35	32, 3, 5, 33, 34	offval2 7620	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑎 ∈ 𝐴 ↦ ((𝐹‘𝑎)𝑅(𝐺‘𝑎))))
36	20, 31, 35	3eqtr4rd 2788	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑁 ∘ 𝑀))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1541   ∈ wcel 2106  Ⅎwnfc 2885  ∀wral 3062  Vcvv 3442  ⟨cop 4584   ↦ cmpt 5180   × cxp 5623   ∘ ccom 5629  ⟶wf 6480  ‘cfv 6484  (class class class)co 7342   ∈ cmpo 7344   ∘_f cof 7598

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pr 5377  ax-un 7655

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-id 5523  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-ov 7345  df-oprab 7346  df-mpo 7347  df-of 7600  df-1st 7904  df-2nd 7905

This theorem is referenced by:  ofpreima  31287  rrvadd  32717