Theorem fcoconst 5808

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 528	. . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → 𝑌 ∈ 𝑋)
2		fconstmpt 4766	. . . 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 5475	. . . . 5 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋⟶V)
6	4, 5	sylib 122	. . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → 𝐹:𝑋⟶V)
7	6	feqmptd 5689	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → 𝐹 = (𝑦 ∈ 𝑋 ↦ (𝐹‘𝑦)))
8		fveq2 5629	. . 3 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
9	1, 3, 7, 8	fmptco 5803	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑌)))
10		fconstmpt 4766	. 2 ⊢ (𝐼 × {(𝐹‘𝑌)}) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑌))
11	9, 10	eqtr4di 2280	1 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹‘𝑌)}))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2799  {csn 3666   ↦ cmpt 4145   × cxp 4717   ∘ ccom 4723   Fn wfn 5313  ⟶wf 5314  ‘cfv 5318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326

This theorem is referenced by: (None)