Theorem oprab2co 8103

Description: Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)

Hypotheses

Ref	Expression
oprab2co.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑅)
oprab2co.2	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑆)
oprab2co.3	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨𝐶, 𝐷⟩)
oprab2co.4	⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑀𝐷))

Assertion

Ref	Expression
oprab2co	⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝑀,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem oprab2co

Step	Hyp	Ref	Expression
1		oprab2co.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑅)
2		oprab2co.2	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑆)
3	1, 2	opelxpd 5704	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝐶, 𝐷⟩ ∈ (𝑅 × 𝑆))
4		oprab2co.3	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨𝐶, 𝐷⟩)
5		oprab2co.4	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑀𝐷))
6		df-ov 7415	. . . . 5 ⊢ (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩)
7	6	a1i 11	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩))
8	7	mpoeq3ia 7492	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑀𝐷)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑀‘⟨𝐶, 𝐷⟩))
9	5, 8	eqtri 2757	. 2 ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑀‘⟨𝐶, 𝐷⟩))
10	3, 4, 9	oprabco 8102	1 ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ⟨cop 4612   × cxp 5663   ∘ ccom 5669   Fn wfn 6535  ‘cfv 6540  (class class class)co 7412   ∈ cmpo 7414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7736

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6493  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7995  df-2nd 7996

This theorem is referenced by: (None)