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Theorem oprab2co 5921
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 4435 . . 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 5597 . . . . 5 (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩)
87a1i 9 . . . 4 ((𝑥𝐴𝑦𝐵) → (𝐶𝑀𝐷) = (𝑀‘⟨𝐶, 𝐷⟩))
98mpt2eq3ia 5652 . . 3 (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑀𝐷)) = (𝑥𝐴, 𝑦𝐵 ↦ (𝑀‘⟨𝐶, 𝐷⟩))
106, 9eqtri 2105 . 2 𝐺 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑀‘⟨𝐶, 𝐷⟩))
114, 5, 10oprabco 5920 1 (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wcel 1436  cop 3428   × cxp 4402  ccom 4408   Fn wfn 4967  cfv 4972  (class class class)co 5594  cmpt2 5596
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-fv 4980  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850
This theorem is referenced by: (None)
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