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Theorem ofc12 7727
Description: Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
ofc12.1 (𝜑𝐴𝑉)
ofc12.2 (𝜑𝐵𝑊)
ofc12.3 (𝜑𝐶𝑋)
Assertion
Ref Expression
ofc12 (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)}))

Proof of Theorem ofc12
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ofc12.1 . . 3 (𝜑𝐴𝑉)
2 ofc12.2 . . . 4 (𝜑𝐵𝑊)
32adantr 480 . . 3 ((𝜑𝑥𝐴) → 𝐵𝑊)
4 ofc12.3 . . . 4 (𝜑𝐶𝑋)
54adantr 480 . . 3 ((𝜑𝑥𝐴) → 𝐶𝑋)
6 fconstmpt 5747 . . . 4 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
76a1i 11 . . 3 (𝜑 → (𝐴 × {𝐵}) = (𝑥𝐴𝐵))
8 fconstmpt 5747 . . . 4 (𝐴 × {𝐶}) = (𝑥𝐴𝐶)
98a1i 11 . . 3 (𝜑 → (𝐴 × {𝐶}) = (𝑥𝐴𝐶))
101, 3, 5, 7, 9offval2 7717 . 2 (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅(𝐴 × {𝐶})) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
11 fconstmpt 5747 . 2 (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥𝐴 ↦ (𝐵𝑅𝐶))
1210, 11eqtr4di 2795 1 (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {csn 4626  cmpt 5225   × cxp 5683  (class class class)co 7431  f cof 7695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697
This theorem is referenced by:  pwsdiagmhm  18844  pwsdiaglmhm  21056  psrlmod  21980  coe1mul2  22272  itg2mulc  25782  dgrmulc  26311  lflvsdi2a  39081  lflvsass  39082  lflsc0N  39084  mendlmod  43201  expgrowth  44354
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