Theorem oprab2co 6273

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		opelxpi 4692	. . 3 ⊢ ((𝐶 ∈ 𝑅 ∧ 𝐷 ∈ 𝑆) → ⟨𝐶, 𝐷⟩ ∈ (𝑅 × 𝑆))
4	1, 2, 3	syl2anc 411	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝐶, 𝐷⟩ ∈ (𝑅 × 𝑆))
5		oprab2co.3	. 2 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨𝐶, 𝐷⟩)
6		oprab2co.4	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑀𝐷))
7		df-ov 5922	. . . . 5 ⊢ (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩)
8	7	a1i 9	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩))
9	8	mpoeq3ia 5984	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑀𝐷)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑀‘⟨𝐶, 𝐷⟩))
10	6, 9	eqtri 2214	. 2 ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑀‘⟨𝐶, 𝐷⟩))
11	4, 5, 10	oprabco 6272	1 ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ⟨cop 3622   × cxp 4658   ∘ ccom 4664   Fn wfn 5250  ‘cfv 5255  (class class class)co 5919   ∈ cmpo 5921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196

This theorem is referenced by: (None)