Theorem fcoconst 5879

Description: Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)

Assertion

Ref	Expression
fcoconst	⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹‘𝑌)}))

Proof of Theorem fcoconst

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 533	. . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → 𝑌 ∈ 𝑋)
2		fconstmpt 4822	. . . 4 ⊢ (𝐼 × {𝑌}) = (𝑥 ∈ 𝐼 ↦ 𝑌)
3	2	a1i 9	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐼 × {𝑌}) = (𝑥 ∈ 𝐼 ↦ 𝑌))
4		simpl 109	. . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → 𝐹 Fn 𝑋)
5		dffn2 5535	. . . . 5 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋⟶V)
6	4, 5	sylib 122	. . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → 𝐹:𝑋⟶V)
7	6	feqmptd 5756	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → 𝐹 = (𝑦 ∈ 𝑋 ↦ (𝐹‘𝑦)))
8		fveq2 5695	. . 3 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
9	1, 3, 7, 8	fmptco 5874	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑌)))
10		fconstmpt 4822	. 2 ⊢ (𝐼 × {(𝐹‘𝑌)}) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑌))
11	9, 10	eqtr4di 2289	1 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹‘𝑌)}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821  {csn 3709   ↦ cmpt 4192   × cxp 4772   ∘ ccom 4778   Fn wfn 5372  ⟶wf 5373  ‘cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385

This theorem is used by: (None)