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Theorem ofc12 5995
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 274 . . 3 ((𝜑𝑥𝐴) → 𝐵𝑊)
4 ofc12.3 . . . 4 (𝜑𝐶𝑋)
54adantr 274 . . 3 ((𝜑𝑥𝐴) → 𝐶𝑋)
6 fconstmpt 4581 . . . 4 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
76a1i 9 . . 3 (𝜑 → (𝐴 × {𝐵}) = (𝑥𝐴𝐵))
8 fconstmpt 4581 . . . 4 (𝐴 × {𝐶}) = (𝑥𝐴𝐶)
98a1i 9 . . 3 (𝜑 → (𝐴 × {𝐶}) = (𝑥𝐴𝐶))
101, 3, 5, 7, 9offval2 5990 . 2 (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝑥𝐴 ↦ (𝐵𝑅𝐶)))
11 fconstmpt 4581 . 2 (𝐴 × {(𝐵𝑅𝐶)}) = (𝑥𝐴 ↦ (𝐵𝑅𝐶))
1210, 11syl6eqr 2188 1 (𝜑 → ((𝐴 × {𝐵}) ∘𝑓 𝑅(𝐴 × {𝐶})) = (𝐴 × {(𝐵𝑅𝐶)}))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  {csn 3522  cmpt 3984   × cxp 4532  (class class class)co 5767  𝑓 cof 5973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-of 5975
This theorem is referenced by: (None)
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