Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fcoconst | Structured version Visualization version GIF version |
Description: Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.) |
Ref | Expression |
---|---|
fcoconst | ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹‘𝑌)})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 788 | . . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → 𝑌 ∈ 𝑋) | |
2 | fconstmpt 5085 | . . . 4 ⊢ (𝐼 × {𝑌}) = (𝑥 ∈ 𝐼 ↦ 𝑌) | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐼 × {𝑌}) = (𝑥 ∈ 𝐼 ↦ 𝑌)) |
4 | simpl 472 | . . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → 𝐹 Fn 𝑋) | |
5 | dffn2 5960 | . . . . 5 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋⟶V) | |
6 | 4, 5 | sylib 207 | . . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → 𝐹:𝑋⟶V) |
7 | 6 | feqmptd 6159 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → 𝐹 = (𝑦 ∈ 𝑋 ↦ (𝐹‘𝑦))) |
8 | fveq2 6103 | . . 3 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌)) | |
9 | 1, 3, 7, 8 | fmptco 6303 | . 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑌))) |
10 | fconstmpt 5085 | . 2 ⊢ (𝐼 × {(𝐹‘𝑌)}) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑌)) | |
11 | 9, 10 | syl6eqr 2662 | 1 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹‘𝑌)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 ↦ cmpt 4643 × cxp 5036 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 |
This theorem is referenced by: s1co 13430 setcmon 16560 pwsco2mhm 17194 pws1 18439 pwsmgp 18441 imasdsf1olem 21988 cvmliftphtlem 30553 cvmlift3lem9 30563 |
Copyright terms: Public domain | W3C validator |