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Theorem oprab2co 5866
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 4403 . . 3 ((𝐶𝑅𝐷𝑆) → ⟨𝐶, 𝐷⟩ ∈ (𝑅 × 𝑆))
41, 2, 3syl2anc 397 . 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 5542 . . . . 5 (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩)
87a1i 9 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩))
98mpt2eq3ia 5597 . . 3 (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑀𝐷)) = (𝑥𝐴, 𝑦𝐵 ↦ (𝑀‘⟨𝐶, 𝐷⟩))
106, 9eqtri 2076 . 2 𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑀‘⟨𝐶, 𝐷⟩))
114, 5, 10oprabco 5865 1 (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  cop 3405   × cxp 4370  ccom 4376   Fn wfn 4924  cfv 4929  (class class class)co 5539  cmpt2 5541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795
This theorem is referenced by: (None)
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