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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 2818 | . . 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 7178 | . . 3 ⊢ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)𝑓) ∈ V | |
3 | 1, 2 | fnmpoi 7757 | . 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 2818 | . . . 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 16957 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑍)𝑓))) |
12 | 11 | fneq1d 6439 | . 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 259 | 1 ⊢ (𝜑 → (〈𝑋, 𝑌〉𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 〈cop 4563 × cxp 5546 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 Basecbs 16471 Hom chom 16564 compcco 16565 compfccomf 16926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-comf 16930 |
This theorem is referenced by: (None) |
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