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Theorem fcoconst 5559
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 504 . . 3 (((𝐹 Fn 𝑋𝑌𝑋) ∧ 𝑥𝐼) → 𝑌𝑋)
2 fconstmpt 4556 . . . 4 (𝐼 × {𝑌}) = (𝑥𝐼𝑌)
32a1i 9 . . 3 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐼 × {𝑌}) = (𝑥𝐼𝑌))
4 simpl 108 . . . . 5 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹 Fn 𝑋)
5 dffn2 5244 . . . . 5 (𝐹 Fn 𝑋𝐹:𝑋⟶V)
64, 5sylib 121 . . . 4 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹:𝑋⟶V)
76feqmptd 5442 . . 3 ((𝐹 Fn 𝑋𝑌𝑋) → 𝐹 = (𝑦𝑋 ↦ (𝐹𝑦)))
8 fveq2 5389 . . 3 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
91, 3, 7, 8fmptco 5554 . 2 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝑥𝐼 ↦ (𝐹𝑌)))
10 fconstmpt 4556 . 2 (𝐼 × {(𝐹𝑌)}) = (𝑥𝐼 ↦ (𝐹𝑌))
119, 10syl6eqr 2168 1 ((𝐹 Fn 𝑋𝑌𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹𝑌)}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  Vcvv 2660  {csn 3497  cmpt 3959   × cxp 4507  ccom 4513   Fn wfn 5088  wf 5089  cfv 5093
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101
This theorem is referenced by: (None)
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