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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 767 | . . 3 ⊢ (((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → 𝑌 ∈ 𝑋) | |
2 | fconstmpt 5614 | . . . 4 ⊢ (𝐼 × {𝑌}) = (𝑥 ∈ 𝐼 ↦ 𝑌) | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐼 × {𝑌}) = (𝑥 ∈ 𝐼 ↦ 𝑌)) |
4 | simpl 485 | . . . . 5 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → 𝐹 Fn 𝑋) | |
5 | dffn2 6516 | . . . . 5 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋⟶V) | |
6 | 4, 5 | sylib 220 | . . . 4 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → 𝐹:𝑋⟶V) |
7 | 6 | feqmptd 6733 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → 𝐹 = (𝑦 ∈ 𝑋 ↦ (𝐹‘𝑦))) |
8 | fveq2 6670 | . . 3 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌)) | |
9 | 1, 3, 7, 8 | fmptco 6891 | . 2 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑌))) |
10 | fconstmpt 5614 | . 2 ⊢ (𝐼 × {(𝐹‘𝑌)}) = (𝑥 ∈ 𝐼 ↦ (𝐹‘𝑌)) | |
11 | 9, 10 | syl6eqr 2874 | 1 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑌 ∈ 𝑋) → (𝐹 ∘ (𝐼 × {𝑌})) = (𝐼 × {(𝐹‘𝑌)})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 ↦ cmpt 5146 × cxp 5553 ∘ ccom 5559 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 |
This theorem is referenced by: s1co 14195 setcmon 17347 pwsco2mhm 17997 smndex1igid 18069 pws1 19366 pwsmgp 19368 imasdsf1olem 22983 cvmliftphtlem 32564 cvmlift3lem9 32574 |
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