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Theorem oprab2co 5965
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 4459 . . 3 ((𝐶𝑅𝐷𝑆) → ⟨𝐶, 𝐷⟩ ∈ (𝑅 × 𝑆))
41, 2, 3syl2anc 403 . 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 5637 . . . . 5 (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩)
87a1i 9 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩))
98mpt2eq3ia 5696 . . 3 (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑀𝐷)) = (𝑥𝐴, 𝑦𝐵 ↦ (𝑀‘⟨𝐶, 𝐷⟩))
106, 9eqtri 2108 . 2 𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑀‘⟨𝐶, 𝐷⟩))
114, 5, 10oprabco 5964 1 (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438  cop 3444   × cxp 4426  ccom 4432   Fn wfn 4997  cfv 5002  (class class class)co 5634  cmpt2 5636
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894
This theorem is referenced by: (None)
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