Theorem fcoconst 5847

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 529	. . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → 𝑌 ∈ 𝑋)
2		fconstmpt 4796	. . . 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 5509	. . . . 5 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋⟶V)
6	4, 5	sylib 122	. . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → 𝐹:𝑋⟶V)
7	6	feqmptd 5729	. . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → 𝐹 = (𝑦 ∈ 𝑋 ↦ (𝐹‘𝑦)))
8		fveq2 5669	. . 3 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
9	1, 3, 7, 8	fmptco 5842	. 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑌)))
10		fconstmpt 4796	. 2 ⊢ (𝐼 × {(𝐹‘𝑌)}) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑌))
11	9, 10	eqtr4di 2283	1 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹‘𝑌)}))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  Vcvv 2812  {csn 3688   ↦ cmpt 4170   × cxp 4746   ∘ ccom 4752   Fn wfn 5346  ⟶wf 5347  ‘cfv 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359

This theorem is referenced by: (None)