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Theorem oprab2co 7149
 Description: Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.)
Hypotheses
Ref Expression
oprab2co.1 ((𝑥𝐴𝑦𝐵) → 𝐶𝑅)
oprab2co.2 ((𝑥𝐴𝑦𝐵) → 𝐷𝑆)
oprab2co.3 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨𝐶, 𝐷⟩)
oprab2co.4 𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑀𝐷))
Assertion
Ref Expression
oprab2co (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
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
StepHypRef Expression
1 oprab2co.1 . . 3 ((𝑥𝐴𝑦𝐵) → 𝐶𝑅)
2 oprab2co.2 . . 3 ((𝑥𝐴𝑦𝐵) → 𝐷𝑆)
3 opelxpi 5072 . . 3 ((𝐶𝑅𝐷𝑆) → ⟨𝐶, 𝐷⟩ ∈ (𝑅 × 𝑆))
41, 2, 3syl2anc 691 . 2 ((𝑥𝐴𝑦𝐵) → ⟨𝐶, 𝐷⟩ ∈ (𝑅 × 𝑆))
5 oprab2co.3 . 2 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨𝐶, 𝐷⟩)
6 oprab2co.4 . . 3 𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑀𝐷))
7 df-ov 6552 . . . . 5 (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩)
87a1i 11 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩))
98mpt2eq3ia 6618 . . 3 (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑀𝐷)) = (𝑥𝐴, 𝑦𝐵 ↦ (𝑀‘⟨𝐶, 𝐷⟩))
106, 9eqtri 2632 . 2 𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑀‘⟨𝐶, 𝐷⟩))
114, 5, 10oprabco 7148 1 (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   × cxp 5036   ∘ ccom 5042   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060 This theorem is referenced by: (None)
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