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Mirrors > Home > MPE Home > Th. List > comffn | Structured version Visualization version GIF version |
Description: The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
comfffn.o | ⊢ 𝑂 = (compf‘𝐶) |
comfffn.b | ⊢ 𝐵 = (Base‘𝐶) |
comffn.h | ⊢ 𝐻 = (Hom ‘𝐶) |
comffn.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
comffn.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
comffn.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
comffn | ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)𝑓)) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)𝑓)) | |
2 | ovex 7188 | . . 3 ⊢ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)𝑓) ∈ V | |
3 | 1, 2 | fnmpoi 7767 | . 2 ⊢ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)𝑓)) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)) |
4 | comfffn.o | . . . 4 ⊢ 𝑂 = (compf‘𝐶) | |
5 | comfffn.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
6 | comffn.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
7 | eqid 2821 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
8 | comffn.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | comffn.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | comffn.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
11 | 4, 5, 6, 7, 8, 9, 10 | comffval 16968 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)𝑓))) |
12 | 11 | fneq1d 6445 | . 2 ⊢ (𝜑 → ((〈𝑋, 𝑌〉𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)) ↔ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)𝑓)) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)))) |
13 | 3, 12 | mpbiri 260 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 〈cop 4572 × cxp 5552 Fn wfn 6349 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 Basecbs 16482 Hom chom 16575 compcco 16576 compfccomf 16937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-comf 16941 |
This theorem is referenced by: (None) |
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