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Theorem ofc12 7707
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 5734 . . . 4 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
76a1i 11 . . 3 (𝜑 → (𝐴 × {𝐵}) = (𝑥𝐴𝐵))
8 fconstmpt 5734 . . . 4 (𝐴 × {𝐶}) = (𝑥𝐴𝐶)
98a1i 11 . . 3 (𝜑 → (𝐴 × {𝐶}) = (𝑥𝐴𝐶))
101, 3, 5, 7, 9offval2 7699 . 2 (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅(𝐴 × {𝐶})) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
11 fconstmpt 5734 . 2 (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥𝐴 ↦ (𝐵𝑅𝐶))
1210, 11eqtr4di 2786 1 (𝜑 → ((𝐴 × {𝐵}) ∘f 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  {csn 4624  cmpt 5225   × cxp 5670  (class class class)co 7414  f cof 7677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679
This theorem is referenced by:  pwsdiagmhm  18776  pwsdiaglmhm  20935  psrlmod  21896  coe1mul2  22181  itg2mulc  25670  dgrmulc  26199  lflvsdi2a  38546  lflvsass  38547  lflsc0N  38549  mendlmod  42611  expgrowth  43766
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