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Theorem oprab2co 7803
 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)
31, 2opelxpd 5566 . 2 ((𝑥𝐴𝑦𝐵) → ⟨𝐶, 𝐷⟩ ∈ (𝑅 × 𝑆))
4 oprab2co.3 . 2 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ ⟨𝐶, 𝐷⟩)
5 oprab2co.4 . . 3 𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑀𝐷))
6 df-ov 7159 . . . . 5 (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩)
76a1i 11 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩))
87mpoeq3ia 7232 . . 3 (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑀𝐷)) = (𝑥𝐴, 𝑦𝐵 ↦ (𝑀‘⟨𝐶, 𝐷⟩))
95, 8eqtri 2781 . 2 𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑀‘⟨𝐶, 𝐷⟩))
103, 4, 9oprabco 7802 1 (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   × cxp 5526   ∘ ccom 5532   Fn wfn 6335  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700 This theorem is referenced by: (None)
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