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Theorem fcoconst 6949
Description: Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
Assertion
Ref Expression
fcoconst ((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹𝑌)}))

Proof of Theorem fcoconst
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 769 . . 3 (((𝐹 Fn 𝑋𝑌𝑋) ∧ 𝑥𝐼) → 𝑌𝑋)
2 fconstmpt 5611 . . . 4 (𝐼 × {𝑌}) = (𝑥𝐼𝑌)
32a1i 11 . . 3 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐼 × {𝑌}) = (𝑥𝐼𝑌))
4 simpl 486 . . . . 5 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹 Fn 𝑋)
5 dffn2 6547 . . . . 5 (𝐹 Fn 𝑋𝐹:𝑋⟶V)
64, 5sylib 221 . . . 4 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹:𝑋⟶V)
76feqmptd 6780 . . 3 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹 = (𝑦𝑋 ↦ (𝐹𝑦)))
8 fveq2 6717 . . 3 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
91, 3, 7, 8fmptco 6944 . 2 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝑥𝐼 ↦ (𝐹𝑌)))
10 fconstmpt 5611 . 2 (𝐼 × {(𝐹𝑌)}) = (𝑥𝐼 ↦ (𝐹𝑌))
119, 10eqtr4di 2796 1 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹𝑌)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  Vcvv 3408  {csn 4541  cmpt 5135   × cxp 5549  ccom 5555   Fn wfn 6375  wf 6376  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388
This theorem is referenced by:  f1ofvswap  7116  s1co  14398  setcmon  17593  pwsco2mhm  18259  smndex1igid  18331  pws1  19634  pwsmgp  19636  imasdsf1olem  23271  cvmliftphtlem  32992  cvmlift3lem9  33002
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