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Theorem ofc12 7073
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 466 . . 3 ((𝜑𝑥𝐴) → 𝐵𝑊)
4 ofc12.3 . . . 4 (𝜑𝐶𝑋)
54adantr 466 . . 3 ((𝜑𝑥𝐴) → 𝐶𝑋)
6 fconstmpt 5302 . . . 4 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
76a1i 11 . . 3 (𝜑 → (𝐴 × {𝐵}) = (𝑥𝐴𝐵))
8 fconstmpt 5302 . . . 4 (𝐴 × {𝐶}) = (𝑥𝐴𝐶)
98a1i 11 . . 3 (𝜑 → (𝐴 × {𝐶}) = (𝑥𝐴𝐶))
101, 3, 5, 7, 9offval2 7065 . 2 (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
11 fconstmpt 5302 . 2 (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥𝐴 ↦ (𝐵𝑅𝐶))
1210, 11syl6eqr 2823 1 (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  {csn 4317  cmpt 4864   × cxp 5248  (class class class)co 6796  𝑓 cof 7046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-of 7048
This theorem is referenced by:  pwsdiagmhm  17577  pwsdiaglmhm  19270  psrlmod  19616  coe1mul2  19854  itg2mulc  23734  dgrmulc  24247  lflvsdi2a  34889  lflvsass  34890  lflsc0N  34892  mendlmod  38289  expgrowth  39060
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